dataset_name
stringclasses 4
values | dataset_version
timestamp[s] | qid
stringlengths 1
5
| queId
stringlengths 32
32
| competition_source_list
sequence | difficulty
stringclasses 5
values | qtype
stringclasses 1
value | problem
stringlengths 6
1.51k
| answer_option_list
list | knowledge_point_routes
sequence | answer_analysis
sequence | answer_value
stringclasses 7
values |
|---|---|---|---|---|---|---|---|---|---|---|---|
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 545 | 9067885c5242404cadc8267d40649401 | [
"其它"
] | 1 | single_choice | How many square numbers and cube numbers are there in the list below? $18$, $27$, $64$, $72$, $81$, $162$, $196$, $324$, $343$, $496$, $529$, $675$, $729$, $784$, $841$, $951$, $1000$ | [
[
{
"aoVal": "A",
"content": "$$9$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$11$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers"
] | [
"Square numbers: $64$, $81$, $196$, $324$, $529$, $729$, $784$, $841$ Cube numbers: $27$, $64$, $343$, $729$, $1000$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 546 | a2d57ba9e3294c80a3cdb257ef309c21 | [] | 1 | single_choice | The ones digit of $$9\times 8\times 7\times 6\times 5\times 4\times 3\times 3\times 4\times 5\times 6\times 7\times 8\times 9$$ is． | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Numbers"
] | [
"Since $$5\\times 4 = 20$$, the ones digit of the given product must be $$0$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 547 | 43ec0f56dfdd4992bf3ab63daffb140f | [
"其它"
] | 0 | single_choice | Which of the following is a composite number? | [
[
{
"aoVal": "A",
"content": "$$59$$ "
}
],
[
{
"aoVal": "B",
"content": "$$61$$ "
}
],
[
{
"aoVal": "C",
"content": "$$63$$ "
}
],
[
{
"aoVal": "D",
"content": "$$67$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers"
] | [
"$63$ is a composite number. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 550 | be87698b58524a658ec8af43503295b4 | [
"其它"
] | 1 | single_choice | How many numbers of the following are divisible by $9$? $$\textasciitilde$$ $452 \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} 387\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash~ ~ ~1057\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash~ ~ ~108\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash~ ~ ~496\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash~ ~ ~1233$ $\textasciitilde$ $\textasciitilde$ | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5$$ "
}
],
[
{
"aoVal": "E",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"$387, 108$, and $1233$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 552 | 999fdde844e945fa82c3ef2a0e0ab0c7 | [] | 1 | single_choice | What is the product of the least common multiple of $$6$$ and $$18$$ and the greatest common factor of $$6$$ and $$18$$? | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$54$$ "
}
],
[
{
"aoVal": "D",
"content": "$$108$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Common Factors and Common Multiples"
] | [
"The least common multiple of $$6$$ and $$18$$ is $$18$$. The greatest common factor of $$6$$ and $$18$$ is $$6$$. Finally, $$6\\times18 = 108$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 553 | 5eee697d3b0c4c718ab9ed27c3bcfb23 | [] | 1 | single_choice | Which of the following numbers are prime numbers? $137$~ ~ ~ ~ ~ ~ ~ ~ ~$151$~ ~ ~ ~ ~ ~ ~ ~ ~$237$~ ~ ~ ~ ~ ~ ~ ~ ~$301$ | [
[
{
"aoVal": "A",
"content": "$137$ and $237$ "
}
],
[
{
"aoVal": "B",
"content": "$137$ and $301$ "
}
],
[
{
"aoVal": "C",
"content": "$237$ and $301$ "
}
],
[
{
"aoVal": "D",
"content": "$137$ and $151$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers"
] | [
"$3$ is the factor of $237$ because $3 \\times79 = 237$; $7$ is the factor of $301$ because $7 \\times 43 = 301$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 554 | a77c39b3628648419e505eff88b4827a | [] | 1 | single_choice | The expression $$1\times 2\times 3\times \cdots \times n$$ has exactly $$128$$ consecutive zeros at the end of its result, then the maximum value of $$n$$ is~\uline{~~~~~~~~~~}~． | [
[
{
"aoVal": "A",
"content": "$$405$$ "
}
],
[
{
"aoVal": "B",
"content": "$$109$$ "
}
],
[
{
"aoVal": "C",
"content": "$$500$$ "
}
],
[
{
"aoVal": "D",
"content": "$$524$$ "
}
],
[
{
"aoVal": "E",
"content": "$$539$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"There are $$128$$ zeros at the end of the result, and the estimated answer is close to $$500$$. $$\\left[ \\frac{500}{5} \\right]+\\left[ \\frac{500}{25} \\right]+\\left[ \\frac{500 }{125} \\right]=100+20+4=124$$, Just add $$4$$ numbers including $$5$$: $$505$$, $$510$$, $$515$$, $$520$$, so the maximum value of $$n$$ that satisfies the condition is $$524$$ . "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 561 | 3b416a9d130d468694f9d9e7660ef8b4 | [
"其它"
] | 1 | single_choice | Think Lab buys $24$ experimental apparatus. But two of the digits of the total price on the invoice are not clear, which only show $\overline{\square 8\square 2}$ dollars. The purchasing agent says the two unclear numbers on the invoice are the same. Given that the single price of this kind of apparatus is an integer, how much is it? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$328$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3$$ "
}
],
[
{
"aoVal": "E",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"The number of total price can be divided by $3$ and $8$. It can be divided by $8$ which means the tens digit must be $3$ or $7$. And when the tens digit is $7$, the number can be divided by $3$. Thus, the total price is $7872$ dollars, and each of them is $328$ dollars. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 562 | 36ef04f664b44136b8414c34bfdf3873 | [] | 2 | single_choice | The hundreds digit of a three-digit number is $$2$$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Applying the Principle of Place Value"
] | [
"$$\\rm Method$$ $$1$$: Let the hundreds, tens, and units digits of the original three-digit number be $$a$$, $$b$$, and $$c$$, respectively. We are given that $$a=c+2$$. The original three-digit number is equal to $$100a+10b+c=100(c+2)+10b+c=101c+10b+200$$. The hundreds, tens, and units digits of the reversed three-digit number are $$c$$, $$b$$, and $$a$$, respectively. This number is equal to $$100c+10b+a=100c+10b+(c+2)=101c+10b+2$$. Subtracting this expression from the expression for the original number, we get $$(101c+10b+200)-(101c+10b+2)=198$$ . Thus, the units digit in the final result is $$8$$. $$\\rm Method$$ $$2$$: The result must hold for any three-digit number with its hundreds digit being $$2$$ more than the units digit. $$301$$ is such a number. Evaluating, we get $$301-103=198$$. Thus, the units digit in the final result is $$8$$. ($$2010$$ AMC $$8$$ Problem, Question \\#$$22$$) "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 563 | 71414ae032e14f0aa8b8eac76ea72baf | [
"其它"
] | 1 | single_choice | I think of a number. When I divide it by $$2$$, the remainder is $$1$$. When I divide it by $$3$$, the remainder is $$2$$. What is the smallest possible value of the number?~\uline{~~~~~~~~~~}~ | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Chinese Remainder Theorem"
] | [
"$$2\\times3-1=5$$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 564 | 5f020e1a25cd41699b22a85414742035 | [] | 1 | single_choice | 375+753+537+357+573+735= ． | [
[
{
"aoVal": "A",
"content": "$$3330$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1000$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3333$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1333$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Applying the Principle of Place Value"
] | [
"$$3+5+7=15$$， $$15+15=30$$， $$30+300+3000=3330$$． "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 565 | 99ae3ee138774a8dbe3b57d1e9eca4cd | [
"其它"
] | 2 | single_choice | Jam has some pieces of candy. He wants to share with some kids. If he shares the candy among $8$ kids equally, there will be $2$ pieces left. If he shares the candy among $9$ kids equally, there will be $3$ pieces left. If he shares the candy among $10$ kids equally, there will be $4$ pieces left. How many pieces of candy are there? | [
[
{
"aoVal": "A",
"content": "$321$ "
}
],
[
{
"aoVal": "B",
"content": "$354$ "
}
],
[
{
"aoVal": "C",
"content": "$720$ "
}
],
[
{
"aoVal": "D",
"content": "$360$ "
}
],
[
{
"aoVal": "E",
"content": "$240$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Chinese Remainder Theorem"
] | [
"The LCF of $8, 9,$ and $10$ is $8\\times9\\times10\\div2=360$. $360-6=354$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 566 | 714239a45f3a404b8bb225d29d7f9f88 | [
"其它"
] | 1 | single_choice | Which of the following number pairs do not contain any divisible relationship? | [
[
{
"aoVal": "A",
"content": "$(12+78), 2$ "
}
],
[
{
"aoVal": "B",
"content": "$(39+61), 3$ "
}
],
[
{
"aoVal": "C",
"content": "$(44+82), 4$ "
}
],
[
{
"aoVal": "D",
"content": "$(25+5100), 5$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"If both numbers in the parentheses are multiple of the smaller number, the number in the parentheses should be divisible by the smaller number. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 568 | 32abb43f945b441aaa75d264ffc39361 | [
"其它"
] | 1 | single_choice | Becky is separating 53 cherries into bag A, bag B and bag C. It is known that there are even number of cherries in both bag A and bag B. Could you tell whether the number of cherries in bag C is even or odd number? | [
[
{
"aoVal": "A",
"content": "Odd "
}
],
[
{
"aoVal": "B",
"content": "Even "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers"
] | [
"Number of cherries in bag C: 53 (odd number) - even - even Number of odd character = 1 Therefore, the number of cherries in bag C is odd. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 570 | 8be0be7f789c43b997ea512be47e1a23 | [
"其它"
] | 1 | single_choice | Think Lab buys $24$ experimental apparatus. But two of the digits of the total price on the invoice are not clear, which only show $\overline{\square 8\square 2}$ dollars. The purchasing agent says the two unclear digits on the invoice are the same. Given that the single price of this kind of apparatus is an integer, how much is it? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$328$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3$$ "
}
],
[
{
"aoVal": "E",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"The number of total price can be divided by $3$ and $8$. It can be divided by $8$ which means the tens digit must be $3$ or $7$. And when the tens digit is $7$, the number can be divided by $3$. Thus, the total price is $7872$ dollars, and each of them is $328$ dollars. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 571 | 4d0261799f564fcb85df22aba9ab70fe | [] | 1 | single_choice | The product of $$4$$ odd numbers is always． | [
[
{
"aoVal": "A",
"content": "even "
}
],
[
{
"aoVal": "B",
"content": "odd "
}
],
[
{
"aoVal": "C",
"content": "less than $$20$$ "
}
],
[
{
"aoVal": "D",
"content": "greater than $$20$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Multiplication Rules of Odd and Even Numbers"
] | [
"The product of odd numbers is always odd. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 572 | 682225b6e5974776bb00ea32874ed249 | [
"其它"
] | 1 | single_choice | There are three whole numbers $A$, $B$ and $C$ ($B\neq1$). $A\times B=21$, $B\times C=57$. $A+B+C=$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$23$$ "
}
],
[
{
"aoVal": "C",
"content": "$$24$$ "
}
],
[
{
"aoVal": "D",
"content": "$$29$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization"
] | [
"$21=3\\times 7$ $57=3\\times 19$ Because $B$ is the factor both number contains, $B=3$ Thus, $A=7$, $C=19$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 573 | 32b4f71f512c4cbbbe395e4fe37c16e4 | [
"其它"
] | 1 | single_choice | Micky and Marcy want to make dumplings. They start with different speeds. There are two kinds of boxes. The smaller one of them could contain $9$ dumplings, and the bigger one could contain $17$ dumplings. Micky works with the smaller one, and Marcy works with the bigger one. What is the smallest number of dumplings that Micky needs to make until they start with a new box respectively at the same time? | [
[
{
"aoVal": "A",
"content": "$$143$$ "
}
],
[
{
"aoVal": "B",
"content": "$$150$$ "
}
],
[
{
"aoVal": "C",
"content": "$$153$$ "
}
],
[
{
"aoVal": "D",
"content": "$$163$$ "
}
],
[
{
"aoVal": "E",
"content": "$$173$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Common Factors and Common Multiples->Common Multiples and Least Common Multiples->Least Common Multiple of Two Numbers"
] | [
"The least common multiple of $9$ and $17$: $9 \\times 17 = 153$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 574 | 32b6afd8cb3c44be8c911a1c85bc3b72 | [] | 1 | single_choice | The $5$-digit number $\overline{2018U}$ is divisible by $9$. What is the remainder when this number is divided by $8$? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"We use the property that the digits of a number must sum to a multiple of $9$ if it is divisible by $9$. This means $2+0+1+8+U$ must be divisible by $9$. The only possible value for U then must be $7$. Since we are looking for the remainder when divided by $8$, we can ignore the thousands. The remainder when $187$ is divided by $8$ is $(\\rm B)$ $3$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 575 | 8be38644dee6484384b66b89cc18c0e1 | [] | 1 | single_choice | $$\sqrt{41^{2} - 9^{2}} + \sqrt{8^{2} + 15^{2}}=$$． | [
[
{
"aoVal": "A",
"content": "$$56$$ "
}
],
[
{
"aoVal": "B",
"content": "$$57$$ "
}
],
[
{
"aoVal": "C",
"content": "$$58$$ "
}
],
[
{
"aoVal": "D",
"content": "$$58$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Basic Applications of Square Numbers"
] | [
"$$\\sqrt{41^{2} - 9^{2}} + \\sqrt{8^{2} + 15^{2}}=\\sqrt{1600} + \\sqrt{289} = 40 + 17 = 57$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 576 | 6cb7bae9da1349f6a6539d819d1de5dc | [] | 1 | single_choice | How many prime numbers are less than $$10$$? | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization->Prime Factorization (Equations)"
] | [
"The prime numbers less than $$10$$ are $$2$$, $$3$$, $$5$$, and $$7$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 579 | 839e166836fb4da990f3922e436d76e6 | [] | 0 | single_choice | A two-digit prime number is still prime when the digits of its first and tenth digits are exchanged. There are such prime numbers. | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"$$11$$，$$13$$，$$17$$，$$31$$，$$37$$，$$71$$，$$73$$，$$79$$，$$97$$． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 580 | 32ccd280d6dc43be996836503fbf4a10 | [] | 2 | single_choice | In a Fibonacci-like sequence $$1,3,4,7,11,18\cdots $$(where each term is the sum of the two previous terms, starting from the third term), what is the remainder when the $$5555^{}\text{th}$$ term is divided by $$5$$？ | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3$$ "
}
],
[
{
"aoVal": "E",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems"
] | [
"Nil "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 582 | a78fcc518cdf4eac805a9742c75b76d5 | [] | 1 | single_choice | The $5$-digit number $\overline{2018U}$ is divisible by $9$. What is the remainder when this number is divided by $8$? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"We use the property that the digits of a number must sum to a multiple of $9$ if it is divisible by $9$. This means $2+0+1+8+U$ must be divisible by $9$. The only possible value for U then must be $7$. Since we are looking for the remainder when divided by $8$, we can ignore the thousands. The remainder when $187$ is divided by $8$ is $(\\rm B)$ $3$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 583 | 75e17d65e3384337b4edaa6d90eef503 | [] | 2 | single_choice | If the product of an even number and an odd number is $$840$$, what is the largest possible value of this odd number? | [
[
{
"aoVal": "A",
"content": "$$21$$ "
}
],
[
{
"aoVal": "B",
"content": "$$35$$ "
}
],
[
{
"aoVal": "C",
"content": "$$105$$ "
}
],
[
{
"aoVal": "D",
"content": "$$420$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Multiplication Rules of Odd and Even Numbers"
] | [
"If the product of an even and an odd number is $$840={2^{3}}\\times105$$, then the largest possible value of the odd number is $$105$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 585 | cc743f7400f744b495eb41f6ec3050e1 | [] | 1 | single_choice | Modify a digit in $$675479$$ so that this six-digit number is divisible by $$25$$. What is the modified six-digit number? | [
[
{
"aoVal": "A",
"content": "$$675480$$ "
}
],
[
{
"aoVal": "B",
"content": "$$675475$$ "
}
],
[
{
"aoVal": "C",
"content": "$$675470$$ "
}
],
[
{
"aoVal": "D",
"content": "$$625479$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders"
] | [
"change the number on ones digit from $$9$$to$$5$$，$$675475$$． "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 587 | b9fa511bbb8f4e688f429dd91a68c321 | [] | 1 | single_choice | Which of the following is a factor of $$380$$? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"Since $$380 =10\\times38$$, $$10$$ is a factor of $$380$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 588 | 51a035336cfa4cd89792b6eeb26402e7 | [] | 1 | single_choice | $$3^{2}+3^{2}+3^{2}+3^{2}=$$． | [
[
{
"aoVal": "A",
"content": "$$4^{2}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6^{2}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12^{2}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$33^{2}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Basic Applications of Square Numbers"
] | [
"$$3^{2}+3^{2}+3^{2}+3^{2}=9+9+9+9=36=6^{2}$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 590 | 908ffabc2d7e463388d04051126de9a0 | [] | 1 | single_choice | $$N$$ is a two$$-$$digit number. When $$N$$ is divided by $$9$$, the remainder is $$1$$. When $$N$$ is divided by $$10$$, the remainder is $$3$$. What is the remainder when $$N$$ is divided by $$11$$? ． | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Characteristics of Remainder "
] | [
"The smallest possible $$N$$ is $$73$$, and $$73 \\div 11\\rm R7$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 592 | ff591f93b84e4c278c48f857934bdf76 | [] | 1 | single_choice | What is the least possible remainder when an even number is divided by $$7$$? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Questions involving Divisions with Remainders"
] | [
"When $$14$$ (or any other even multiple of $$7$$) is divided by $$7$$, the remainder is $$0$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 594 | 63b475d8986743d9a318d3e19919ac30 | [] | 1 | single_choice | $$18$$ apples are equally given to $$9$$ kids. How many apples can every kid get? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$2$$ "
}
],
[
{
"aoVal": "E",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders"
] | [
"$$18\\div9=2$$ $$18=2+2+2+2+2+2+2+2+2$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 596 | 5a9f067862ad466b8516c36bce658b8b | [] | 0 | single_choice | Which of the following numbers is not prime? | [
[
{
"aoVal": "A",
"content": "$$13 $$ "
}
],
[
{
"aoVal": "B",
"content": "$$19 $$ "
}
],
[
{
"aoVal": "C",
"content": "$$89$$ "
}
],
[
{
"aoVal": "D",
"content": "$$93$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers->Determining Prime and Composite Numbers"
] | [
"$$93=31\\times 3$$ , its factors are $1$，$31$，$3$，$93$． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 599 | 716675a0e9af4299962aefdaa69aa99d | [
"其它"
] | 1 | single_choice | If $n$ and $m$ are integers and $n^{2}+m^{2}$ is even, which of the following is impossible? (2014 AMC 8 Problem, Question \#13) | [
[
{
"aoVal": "A",
"content": "$n$ and $m$ are even "
}
],
[
{
"aoVal": "B",
"content": "$n$ and $m$ are odd "
}
],
[
{
"aoVal": "C",
"content": "$n+m$ is even "
}
],
[
{
"aoVal": "D",
"content": "$n+m$ is odd "
}
],
[
{
"aoVal": "E",
"content": "none of these are impossible "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"The question asks which one is impossible, all we need to do is find one possible way that the others are possible. After trying, when $n$ and $m$ are both even or odd, the calculation works, so $D$ is not correct. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 604 | fabee4e3dd6240d3b1e6b55192cb4ae8 | [
"其它"
] | 0 | single_choice | Which of the following is not a factor of $48$? | [
[
{
"aoVal": "A",
"content": "$$8$$ "
}
],
[
{
"aoVal": "B",
"content": "$$12$$ "
}
],
[
{
"aoVal": "C",
"content": "$$14$$ "
}
],
[
{
"aoVal": "D",
"content": "$$16$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"$14$ can not divide $48$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 606 | 5634385348724933ae49b4983a89c14a | [] | 1 | single_choice | Find the value for lcm $$\left[ 12,18\right]$$. | [
[
{
"aoVal": "A",
"content": "$$36$$ "
}
],
[
{
"aoVal": "B",
"content": "$$24$$ "
}
],
[
{
"aoVal": "C",
"content": "$$72$$ "
}
],
[
{
"aoVal": "D",
"content": "$$48$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Common Factors and Common Multiples"
] | [
"List the prime factorization for $$12$$ and $$18$$ first. $$12=2^{2}\\times3$$ and $$18=3^{2}\\times2$$. The largest exponent for $$2$$ is $$2$$, and the larger exponent for $$3$$ is $$2$$, thus the least common multiple for $$12$$ and $$18$$ is $$2^{2}\\times3^{2}=36$$. We choose $$\\text{A}$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 608 | 684f922efc664011860b8f69d3faec5e | [] | 1 | single_choice | If $$a=1$$, $$b=2$$, and $$c=3$$, then $$\overline{abc}=123$$. Find the value of $$m$$ according to this rule. $$\overline{m21}=8\times \overline{m9}+3m$$. | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Applying the Principle of Place Value"
] | [
"$$\\begin{eqnarray}\\overline{m21}\\&=\\&8\\times \\overline{m9}+3m\\textbackslash\\textbackslash{} 100m+21\\&=\\&80m+72+3m\\textbackslash\\textbackslash{} 17m\\&=\\&51\\textbackslash\\textbackslash{} m\\&=\\&3.\\end{eqnarray}$$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 610 | 375e4ea1c2c64564952bff0f8058c931 | [] | 1 | single_choice | Answer the question below: If $$a$$, $$b$$ are prime numbers, and $$3a+7b=41$$, then $$a+b=$$． | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers->Applying Special Prime Numbers"
] | [
"Based on the laws relating to the parity in addition and multiplication, either $$a$$ or $$b$$ must be $$2$$. If $$a = 2$$, then $$b = 5$$, and $$a + b = 7$$; if $$b = 2$$, then $$a = 9$$; $$9$$ is not a prime number, which doesn\\textquotesingle t match the conditions in the question. Therefore, we choose B. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 611 | 7a94aab4d24747c69907e93f9ab1c856 | [
"其它"
] | 1 | single_choice | A kind of water plant grows so fast that it doubles every day. If one plant is put into the pond on the first day, it will turn to two plants on the second day, and on the $26$\textsuperscript{th} days, they can fill the pond. If $8$ water plants are put into the pond on the first day, how many days will it take to fill the pond? | [
[
{
"aoVal": "A",
"content": "$$21$$ "
}
],
[
{
"aoVal": "B",
"content": "$$22$$ "
}
],
[
{
"aoVal": "C",
"content": "$$23$$ "
}
],
[
{
"aoVal": "D",
"content": "$$24$$ "
}
],
[
{
"aoVal": "E",
"content": "$$25$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"It takes three days for one plant to turn to $8$ plants. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 614 | 5ac2b47d84bb474ca9ea93b9dfdd71a7 | [
"其它"
] | 1 | single_choice | A number has the same digit in its tens place and its hundredth place. How many times greater is the value of the digit in the tens place than the value of the digit in the hundredth place? | [
[
{
"aoVal": "A",
"content": "$0.1$ "
}
],
[
{
"aoVal": "B",
"content": "$100$ "
}
],
[
{
"aoVal": "C",
"content": "$1000$ "
}
],
[
{
"aoVal": "D",
"content": "$10,000$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases"
] | [
"Let the digit be $d$ $10d\\div 0.01d=1000$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 615 | ecdb187f8f5446a4b97760216e1c206e | [] | 1 | single_choice | If an equilateral triangle has in-teger sides, its perimeter \emph{cannot} be． | [
[
{
"aoVal": "A",
"content": "$$915$$ "
}
],
[
{
"aoVal": "B",
"content": "$$615$$ "
}
],
[
{
"aoVal": "C",
"content": "$$315$$ "
}
],
[
{
"aoVal": "D",
"content": "$$115$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Applying the Properties of Dividing without Remainders"
] | [
"The perimeter of an equilateral $$\\triangle $$ with integer sides is divisible by $$3$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 616 | b0dcbc585b31462aafe6c6d19e0604d7 | [] | 1 | single_choice | How many different primes are in the prime factorisation of $$2016$$? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization->Prime Factorization (Equations)"
] | [
"$$2016=2\\times2\\times2\\times2\\times2\\times3\\times3\\times7$$; there are $$3$$ different primes. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 617 | 51d7355e6f1b4cc8903463abcfdabdbf | [] | 1 | single_choice | If $$1\times2\times3\times4\times5\times6\times7\times8\times$$$$9\times10\times11\times12\times13\times$$$$14\times15 = 1307 674 368000$$, how many times does the digit "$$0$$" appear in the product $$10\times20\times30\times40\times50\times60\times$$$$70\times80\times90\times100\times110\times120\times130\times140\times150$$? | [
[
{
"aoVal": "A",
"content": "$$15$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$19$$ "
}
],
[
{
"aoVal": "D",
"content": "$$20$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization->Applying Prime Factorization"
] | [
"If $$1\\times2\\times3\\times4\\times5\\times6\\times7\\times$$$$8\\times9\\times10\\times11\\times12\\times13\\times14\\times15 = $$$$1307 674 368000$$, and we multiply each of these $$15$$ numbers by $$10$$, the new product will have an additional $$15$$ zeroes, and $$15+4=19$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 618 | 760c60c12063487c846f1fadc8010ba0 | [] | 1 | single_choice | The $5$-digit number $\overline{2018U}$ is divisible by $9$. What is the remainder when this number is divided by $8$? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"We use the property that the digits of a number must sum to a multiple of $9$ if it is divisible by $9$. This means $2+0+1+8+U$ must be divisible by $9$. The only possible value for U then must be $7$. Since we are looking for the remainder when divided by $8$, we can ignore the thousands. The remainder when $187$ is divided by $8$ is $(\\rm B)$ $3$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 624 | 4493dee2b75046bc9ce0378b59f92a2f | [] | 1 | single_choice | If add $$1994$$ to any odd number, the sum will always be． | [
[
{
"aoVal": "A",
"content": "odd "
}
],
[
{
"aoVal": "B",
"content": "even "
}
],
[
{
"aoVal": "C",
"content": "$$1995$$ "
}
],
[
{
"aoVal": "D",
"content": "prime "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Addition and Subtraction Rules of Odd and Even Numbers"
] | [
"even number $$+$$ odd number $$=$$ odd number. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 627 | 5adba830ff3b46ec92b30e403d6aa2cc | [
"其它"
] | 1 | single_choice | Kate and Kerry are two dogs. Every $3$-hour, Kate goes to the dog park. Every $2$-hour, Kerry gos to the dog park. Today, Kate and Kerry first meet at $10:00\text{am}$. When will they meet each other again? | [
[
{
"aoVal": "A",
"content": "$12:00\\text{pm}$ "
}
],
[
{
"aoVal": "B",
"content": "$4:00\\text{pm}$ "
}
],
[
{
"aoVal": "C",
"content": "$6:00\\text{pm}$ "
}
],
[
{
"aoVal": "D",
"content": "$10:00\\text{pm}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Common Factors and Common Multiples->Common Multiples and Least Common Multiples->Least Common Multiple of Two Numbers"
] | [
"$LCM[2,3]=6$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 630 | a7b5f72c56164004b54c43e5b88481ea | [] | 1 | single_choice | The characteristic of numbers that are divisible by $$4$$ is． | [
[
{
"aoVal": "A",
"content": "Last digit is divisible by $$4$$. "
}
],
[
{
"aoVal": "B",
"content": "Last two digits are divisible by $$4$$. "
}
],
[
{
"aoVal": "C",
"content": "Last three digits are divisible by $$4$$. "
}
],
[
{
"aoVal": "D",
"content": "The sum of digits is divisible by $$4$$. "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders"
] | [
"$$1$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 631 | 490d5358ce7046a4b9b81cb2fe5a90fa | [] | 1 | single_choice | $$\sqrt {2\times 4\times 8}\times \sqrt {8\times 8}=$$． | [
[
{
"aoVal": "A",
"content": "$$64$$ "
}
],
[
{
"aoVal": "B",
"content": "$$32$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Basic Applications of Square Numbers"
] | [
"$$\\sqrt {2\\times 4\\times 8}\\times \\sqrt {8\\times 8}=8\\times 8=64$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 634 | 37b5bc085e0d460392df19ab9a41c416 | [] | 1 | single_choice | Which of the following numbers is not the square of a whole number? | [
[
{
"aoVal": "A",
"content": "$$100$$ "
}
],
[
{
"aoVal": "B",
"content": "$$144$$ "
}
],
[
{
"aoVal": "C",
"content": "$$196$$ "
}
],
[
{
"aoVal": "D",
"content": "$$200$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Basic Applications of Square Numbers"
] | [
"All choices except $$200$$ are perfect squares since: $$100=10^{2}$$ $$144=12^{2}$$ $$196=14^{2}$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 637 | 6d0a18cdd59d4557a14f76ecd508eba5 | [
"其它"
] | 1 | single_choice | How many factors does $36$ have? | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"$1, 2, 3, 4, 6, 9, 12, 18, 36$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 639 | 5f74f620bffe44f380949499e519b587 | [] | 1 | single_choice | When from any three-digit number we subtract that number written backwards, the difference will always be a number that is divisible by:． | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Casting Out Nines"
] | [
"No matter how we write the three-digit number, the sum of its three digits is always the same. Therefore, the remainders of the two numbers divided by $$9$$ are also the same. The difference will always be a number that is divisible by $$9$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 640 | 7ac01a3bb1194ff7b78925d0d02f7485 | [] | 0 | single_choice | Which of the following has an odd number of whole-number factors? | [
[
{
"aoVal": "A",
"content": "$$47$$ "
}
],
[
{
"aoVal": "B",
"content": "$$48$$ "
}
],
[
{
"aoVal": "C",
"content": "$$49$$ "
}
],
[
{
"aoVal": "D",
"content": "$$50$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Understanding Odd and Even Numbers"
] | [
"The whole-number factors of $$49$$ are $$1$$, $$7$$, and $$49$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 642 | 5f8027a5003742e692777d0509b09153 | [
"其它"
] | 1 | single_choice | How many positive integer factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely $1,2,3,4,6$, and 12 .) (2020 AMC 8 Problem, Question \#17) | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"Since $2020=2^{2} \\cdot 5 \\cdot 101$, we can simply list its factors: $$ 1,2,4,5,10,20,101,202,404,505,1010,2020 . $$ There are 12 of these; only $1,2,4,5,101$ (i.e. 5 of them) don\\textquotesingle t have over 3 factors, so the remaining $12-5=(\\text{B}) 7$ factors have more than 3 factors. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 643 | e3aaeda909b642009297690f00b114b3 | [
"其它"
] | 2 | single_choice | From 202 to 2020 (including these two numbers), how many multiples of 9 are there? | [
[
{
"aoVal": "A",
"content": "$$201$$ "
}
],
[
{
"aoVal": "B",
"content": "$$202$$ "
}
],
[
{
"aoVal": "C",
"content": "$$101$$ "
}
],
[
{
"aoVal": "D",
"content": "$$102$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"$[\\frac{2020}{9}]-[\\frac{201}{9}]=224-22=202$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 651 | d139cdcaffda4f6e921a448ed14b69ec | [
"其它"
] | 2 | single_choice | Amy runs a lap around the track in $4$ minutes and Pawel in $5$ minutes. Amy and Pawel start to run around the track at the same time. After how many minutes will the boys meet at the starting point again? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$20$$ "
}
],
[
{
"aoVal": "E",
"content": "It depends on the distance around the track "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"The answer is the least common multiple of $4$ and $5$, which is $20$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 654 | df12c8ec69db488da6ffcaf5e0b9fa02 | [] | 1 | single_choice | How many positive factors of $$144$$ are also multiples of $$4$$? | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"$$144={{2}^{4}}\\times {{3}^{2}}=4\\times2^{2}\\times {{3}^{2}}$$. Among them, there are $$\\left( 2+1 \\right)\\times (2+1)=9$$ factors which has $$4$$ as its factor. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 655 | 4953f555328044d6bf88c2402e0e952b | [
"其它"
] | 1 | single_choice | John, Emily and Nick want to buy some apples from Walmart where the apples are sold in pack of four. If they buy six packs and share the apples evenly, how many apples will each one of them get? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"$4\\times 6=24$; $24=3\\times 8$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 656 | 56a8b35b148f432795b814652a9d54c0 | [] | 1 | single_choice | Each of the following results in an even number except． | [
[
{
"aoVal": "A",
"content": "$$952+136$$ "
}
],
[
{
"aoVal": "B",
"content": "$$952-136$$ "
}
],
[
{
"aoVal": "C",
"content": "$$952\\div 136$$ "
}
],
[
{
"aoVal": "D",
"content": "$$952\\times 136$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Odd and Even Applications"
] | [
"The sum, difference, and product of even numbers are even, but $$952\\div 136=7$$ is odd. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 658 | ba36917be0014d168e2984e4500780b7 | [] | 1 | single_choice | What is the smallest prime number greater than $$47$$? | [
[
{
"aoVal": "A",
"content": "$$59$$ "
}
],
[
{
"aoVal": "B",
"content": "$$57$$ "
}
],
[
{
"aoVal": "C",
"content": "$$53$$ "
}
],
[
{
"aoVal": "D",
"content": "$$51$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers->Determining Prime and Composite Numbers"
] | [
"$$49 =7\\times7$$; $$51 = 3\\times17$$. The first prime is $$53$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 659 | 88a00507342d47c1b0e19a72002f66af | [
"其它"
] | 1 | single_choice | For a positive integer $n$, the factorial notation $n!$ repressents the product of the integer from $n$ to $1$, (For example, $6!=6\times5\times4\times3\times2\times1$.) What value of $N$ satisfies the following equation? $$6!\times8!=8\times N!$$ | [
[
{
"aoVal": "A",
"content": "$$9$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$11$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
],
[
{
"aoVal": "E",
"content": "$$13$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples"
] | [
"$6!=10\\times9\\times8$, $$6!\\times8!=8\\times 10!$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 663 | 84092477536143fb9d381ed652715e82 | [] | 1 | single_choice | Find the smallest multiple of $$7$$ that is greater than or equal to $$20 $$. | [
[
{
"aoVal": "A",
"content": "$$20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$21$$ "
}
],
[
{
"aoVal": "C",
"content": "$$22$$ "
}
],
[
{
"aoVal": "D",
"content": "$$28$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"$$2\\times7=14$$, $$3\\times7=21$$, since $$14$$ is smaller than $$20$$ and $$21$$ is the closest multiple of $$7$$ and greater than $$20$$, we choose $$\\text B$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 664 | 765fa02e95c34931aefa0929857642ee | [
"其它"
] | 1 | single_choice | There are three whole number $A$, $B$, $C$. $A\times B=77$, $B\times C=132$. $A+B+C=$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$30$$ "
}
],
[
{
"aoVal": "B",
"content": "$$31$$ "
}
],
[
{
"aoVal": "C",
"content": "$$32$$ "
}
],
[
{
"aoVal": "D",
"content": "$$33$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization"
] | [
"$77=7\\times 11$ $132=2\\times 2\\times 3\\times 11$ Because $B$ is the factor both number contains, $B=11$ Thus, $A=7$, $C=12$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 670 | 68d281debfa344ca90d42e8daaf20953 | [
"其它"
] | 0 | single_choice | Which of the following number is a multiple of $9$? | [
[
{
"aoVal": "A",
"content": "$$234$$ "
}
],
[
{
"aoVal": "B",
"content": "$$136$$ "
}
],
[
{
"aoVal": "C",
"content": "$256$ "
}
],
[
{
"aoVal": "D",
"content": "$$418$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders"
] | [
"$2+3+4=9$, so $234$ is divisible by 9. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 673 | 90f85ce1e78b4a008e9b93aea2c4b3d7 | [
"其它"
] | 2 | single_choice | The smallest number greater than $2$ that leaves a remainder of $2$ when divided by $3,4,5$, or $6$ lies between what numbers? (2012 AMC 8 Problem, Question \# 15) | [
[
{
"aoVal": "A",
"content": "$40$ and $50$ "
}
],
[
{
"aoVal": "B",
"content": "$51$ and $55$ "
}
],
[
{
"aoVal": "C",
"content": "$56$ and $60$ "
}
],
[
{
"aoVal": "D",
"content": "$61$ and $65$ "
}
],
[
{
"aoVal": "E",
"content": "$66$ and $99$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"First, list number that leaves a remainder of $2$ when divided by $6$: $8,14,20,26,32,38,44,50,56,62\\cdots $. Meanwhile, divide each number with $5$ to see it the number meets the condition. If the number divided by $5$ leaves a remainder of $2$, divide it by $4$ see if it still works. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 674 | d14f4b95c61d47f5a993a06a2c5c3422 | [] | 1 | single_choice | The product of $$2$$ odd numbers is always. | [
[
{
"aoVal": "A",
"content": "divisible by $$3$$ "
}
],
[
{
"aoVal": "B",
"content": "odd "
}
],
[
{
"aoVal": "C",
"content": "prime "
}
],
[
{
"aoVal": "D",
"content": "even "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Multiplication Rules of Odd and Even Numbers"
] | [
"The product of $$2$$ odd numbers, such as $$5\\times7=35$$, is always odd. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 675 | 40ecd025d4234849a565eb6f20a21835 | [
"其它"
] | 1 | single_choice | What does the "3"~in 1234 mean? | [
[
{
"aoVal": "A",
"content": "3 hundreds "
}
],
[
{
"aoVal": "B",
"content": "3 tens "
}
],
[
{
"aoVal": "C",
"content": "3 thousands "
}
],
[
{
"aoVal": "D",
"content": "3 ones "
}
],
[
{
"aoVal": "E",
"content": "3 millions "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Numbers->Understanding Numbers and Digits"
] | [
"3 tens "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 677 | 454bba4a33824efab618378ac8426ce0 | [] | 0 | single_choice | A two-digit prime number is still prime when the digits of its first and tenth digits are exchanged. There are such prime numbers. | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"$$11$$，$$13$$，$$17$$，$$31$$，$$37$$，$$71$$，$$73$$，$$79$$，$$97$$． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 681 | b5b5215a73eb4d8e93d34a0e05f7a435 | [] | 1 | single_choice | Fill in the blanks below with the largest possible numbers: ~\uline{~~~~~~~~~~}~$$\div16 = 5\rm R$$~\uline{~~~~~~~~~~}~. ~\uline{~~~~~~~~~~}~$$\div16 = 5\rm$$ $\cdots\cdots$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "105, 15 "
}
],
[
{
"aoVal": "B",
"content": "95, 15 "
}
],
[
{
"aoVal": "C",
"content": "88, 13 "
}
],
[
{
"aoVal": "D",
"content": "64, 16 "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Questions involving Divisions with Remainders"
] | [
"The largest possible remainder should be at least $$1$$ smaller than the divisor. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 684 | 8c68c724c9934a358b17bc7a8403f6ea | [] | 1 | single_choice | What is the missing number in the box? $$7063000 =7000000+\boxed{?}+3000$$ | [
[
{
"aoVal": "A",
"content": "$$600$$ ones "
}
],
[
{
"aoVal": "B",
"content": "$$600$$ tens "
}
],
[
{
"aoVal": "C",
"content": "$$600$$ hundreds "
}
],
[
{
"aoVal": "D",
"content": "$$600$$ thousands "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Numbers"
] | [
"$600$ hundreds has the same value as $600\\times100=60000$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 687 | 9ec72666d4f5427bb1ae52f1facdc562 | [] | 0 | single_choice | Which of the following cases can\textquotesingle t make an even number? | [
[
{
"aoVal": "A",
"content": "An odd number$$+$$An even number "
}
],
[
{
"aoVal": "B",
"content": "An even number$$+$$An even number "
}
],
[
{
"aoVal": "C",
"content": "An odd number$$+$$An odd number "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Multiplication Rules of Odd and Even Numbers"
] | [
"The sum of an odd number and an even number is always an odd~ number. The sum of two numbers which are both even or both odd is always an even number. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 688 | 5b75f53e71c64811acdd402bbb5ac797 | [] | 2 | single_choice | Divide 8 numbers $$15$$、$$18$$、$$21$$、$$22$$、$$42$$、$$44$$、$$50$$ and $$60$$ into two groups with 4 numbers in each group to make the product of numbers in each group the same, so the two groups are ． | [
[
{
"aoVal": "A",
"content": "（$$15$$, $$22$$, $$21$$, $$60$$），（$$18$$, $$44$$, $$42$$, $$50$$） "
}
],
[
{
"aoVal": "B",
"content": "（$$15$$, $$42$$, $$44$$, $$60$$），（$$18$$, $$22$$, $$21$$, $$50$$） "
}
],
[
{
"aoVal": "C",
"content": "（$$15$$, $$44$$, $$21$$, $$60$$），（$$18$$, $$22$$, $$42$$, $$50$$） "
}
],
[
{
"aoVal": "D",
"content": "（$$15$$, $$44$$, $$21$$, $$50$$），（$$18$$, $$22$$, $$42$$, $$60$$） "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization->Applying Prime Factorization->Finding Factors Given the Product"
] | [
"omitted "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 689 | 456df3204842430d8a3e9cfae8938155 | [
"其它"
] | 1 | single_choice | A kind of water plant grows so fast that it doubles every day. If one plant is put into the pond on the first day, two plants will grow on the second day, and on the twenty-sixth days, they will just fill the pond. If eight water plants are put into the pond on the first day, how many days it will take to fill the pond? | [
[
{
"aoVal": "A",
"content": "$$21$$ "
}
],
[
{
"aoVal": "B",
"content": "$$22$$ "
}
],
[
{
"aoVal": "C",
"content": "$$23$$ "
}
],
[
{
"aoVal": "D",
"content": "$$24$$ "
}
],
[
{
"aoVal": "E",
"content": "$$25$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"It takes three days for one plant to develop to eight plants. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 691 | 49c5137498984ceb90c520f6e5483bc7 | [] | 1 | single_choice | What is the remainder when $$12+34+56+89+90$$ is divided by $$10$$? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Questions involving Divisions with Remainders"
] | [
"Add ones digits, then divide by $$10$$: $$2+4+6+9+0=21$$;~$$21\\div 10=2\\text{R}1$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 693 | 570e665b4c784481b3bd0cc061cdd7d2 | [] | 1 | single_choice | Only one of the following four numbers is a perfect square. Which one is it? | [
[
{
"aoVal": "A",
"content": "$$76186$$ "
}
],
[
{
"aoVal": "B",
"content": "$$750235$$ "
}
],
[
{
"aoVal": "C",
"content": "$$921438$$ "
}
],
[
{
"aoVal": "D",
"content": "$$2660161$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Basic Applications of Square Numbers"
] | [
"$$\\text{A}$$: $$76186\\div2=28093$$. But $$28093$$ is not divisible by $$2$$. So, $$76186$$ is divisible by $$2$$ but not $$4$$, $$76186$$ is not a perfect square. $$\\text{B}$$: $$750235\\div5=150047$$. But $$150047$$ is not divisible by $$5$$. So, $$750235$$ is divisible by $$5$$ but not $$25$$, $$750235$$ is not a perfect square. $$\\text{C}$$: $$921438\\div2=460719$$. But $$460719$$ is not divisible by $$2$$. So, $$921438$$ is divisible by $$2$$ but not $$4$$, $$921438$$ is not a perfect square. Thus, answer must be $$\\text{D}$$. In fact, $$2660161=7^{2}\\times233^{2}$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 695 | ccc29422593e4417a518581693ae5b1c | [
"其它"
] | 1 | single_choice | What is the sum of the prime factors of $2010$? (2010 AMC 8 Problems, Question \#14) | [
[
{
"aoVal": "A",
"content": "$$67$$ "
}
],
[
{
"aoVal": "B",
"content": "$$75$$ "
}
],
[
{
"aoVal": "C",
"content": "$$77$$ "
}
],
[
{
"aoVal": "D",
"content": "$$201$$ "
}
],
[
{
"aoVal": "E",
"content": "$$210$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers"
] | [
"First, we must find the prime factorization of $2010.2010=2 \\cdot 3 \\cdot 5 \\cdot 67$. We add the factors up to get $(\\mathbf{C}) 77$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 697 | 7213f4b7283848b9a9aada3ddac57686 | [] | 2 | single_choice | The whole numbers from $$1$$ to $$2016$$ inclusive are written on a blackboard. Moritz underlines all the multiples of two in red, all the multiples of three in blue and all the multiples of four in green. How many numbers does Moritz underline exactly twice? | [
[
{
"aoVal": "A",
"content": "$$1008$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1004$$ "
}
],
[
{
"aoVal": "C",
"content": "$$504$$ "
}
],
[
{
"aoVal": "D",
"content": "$$336$$ "
}
],
[
{
"aoVal": "E",
"content": "$$168$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"There is no number that is both a multiple of three and a multiple of four without also being a multiple of two. Hence, the numbers underlined exactly twice are those that are a multiple of two and of three but not of four and those that are a multiple of two and four but not of three. The first set of numbers consists of the set of odd multiples of six. Since $$2016 \\div 6 = 336$$, there are $$336$$ multiples of $$6$$ in the list of numbers and hence $$336 \\div 2 = 168$$ odd multiples of six that would be underlined in red and blue but not green. The second set of numbers consists of two out of every three multiples of four and, since $$2016 \\div 4 = 504$$, there are $$3\\times 504 = 336 $$ numbers that would be underlined in red and green but not blue. Hence there are $$168+ 336 = 504$$ numbers that Moritz would underline exactly twice. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 699 | 571af844dc1c490eb785699428b35c6c | [] | 1 | single_choice | Which of the following is a prime number? | [
[
{
"aoVal": "A",
"content": "$$38$$ "
}
],
[
{
"aoVal": "B",
"content": "$$45$$ "
}
],
[
{
"aoVal": "C",
"content": "$$53$$ "
}
],
[
{
"aoVal": "D",
"content": "$$57$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers->Determining Prime and Composite Numbers"
] | [
"We note that the other options are not prime numbers because: $38 = 2 \\times 19$ $45 = 4 \\times 9$ $57 = 3 \\times 19$ Hence by the process of elimination, $53$ is a prime number. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 700 | 6d8d8cf0a281433dbf23cfc2674320f2 | [] | 1 | single_choice | A student wrote down a natural number. When she divided the number by $$9$$, the remainder was $$7$$. What is the~~remainder when twice that number is divided by $$9$$? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Characteristics of Remainder "
] | [
"The remainder of $$A$$ $$\\div9$$ is $$7$$, and $$2A=A+A$$. Therefore the remainder of $$2A\\div9$$ is $$7+7=14$$. $$14= 9+5$$, therefore the remainder is $$5$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 701 | f1b28656354347dd871ba79d11525dd9 | [] | 1 | single_choice | is a factor of $$100110011001$$. | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"$$100110011001$$ is divisible by $$3$$, since the sum of its digits is $6$ which is divisible by $3$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 702 | 571eafff8fb2450087a8eef4928cd018 | [] | 1 | single_choice | The product of any $$3$$-digit whole number and any $$2$$-digit whole number can contain at most how many digits? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Numbers"
] | [
"The largest product of a $$3$$-digit number and a $$2$$-digit number is $$999\\times99 = 98901$$. That product has $$5$$ digits. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 704 | 648b4f56ce59455198880d8914b485c0 | [] | 2 | single_choice | What is the greatest natural number $$n$$ such that $$n+27$$ and $$n-62$$ are squares of natural number? | [
[
{
"aoVal": "A",
"content": "$$598$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1598$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3998$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1998$$ "
}
],
[
{
"aoVal": "E",
"content": "Such a number does not exist "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Questions involving Square Numbers"
] | [
"Let $a^{2} = n+27$ and $b^{2} = n-62$. Taking the difference: $a^{2}-b^{2} = (a+b)(a-b) = n+27-(n-62) = 89$ Since $89$ is prime, that means $a+b=89$ and $a-b=1$, which means $a=45$ and $b=44$. That means $n=a^{2}-27 = 45^{2}-27 = 2025-27=1998$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 706 | ba5f15593e6f4c7aa8ce42e1f7dfcd04 | [] | 1 | single_choice | If I write all the whole numbers from $$1$$ to $$100$$ in words, how many times will I write the letter V? | [
[
{
"aoVal": "A",
"content": "$$19$$ "
}
],
[
{
"aoVal": "B",
"content": "$$29$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30$$ "
}
],
[
{
"aoVal": "D",
"content": "$$31$$ "
}
],
[
{
"aoVal": "E",
"content": "$$32$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Applying the Principle of Place Value"
] | [
"There are $$10$$ sevens in the unit digit, $$10$$ seventys, $$9$$ fives in the units (as fifteen is not fiveteen) plus eleven and twelve. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 708 | 9a4528f59c2643a3936b64f8f8823092 | [] | 1 | single_choice | How many multiples of $5$ are there from $1$ to $50$? | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$15$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"$5$, $10$, $15$, $20$, $25$, $30$, $35$, $40$, $45$, $50$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 715 | 69259a06ea8b4ca0a5a83423ffef5774 | [] | 1 | single_choice | Which of the following prime factorizations below is correct? | [
[
{
"aoVal": "A",
"content": "$$97=1\\times97$$ "
}
],
[
{
"aoVal": "B",
"content": "$$85=5\\times 17$$ "
}
],
[
{
"aoVal": "C",
"content": "$$64=8\\times 8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$52=1\\times 2\\times 2\\times 13$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization"
] | [
"①A prime number is a natural number greater than $$1$$ that has no other factors than $$1$$ and itself. ②Decomposing a composite number into the product of several prime factors is called decomposing prime factors. ③$$1$$ is neither a prime number nor a composite number. So $$\\text{D}$$ is not correct. The $$\\text{A}$$ option is also incorrect. $$8$$ is not a prime number, so $$\\text{C}$$ is incorrect. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 716 | 5bae882a875b414ba8614df071ff4265 | [] | 1 | single_choice | Which of the followings is the least common multiple for $$25$$ and $$125$$? | [
[
{
"aoVal": "A",
"content": "$$25$$ "
}
],
[
{
"aoVal": "B",
"content": "$$125$$ "
}
],
[
{
"aoVal": "C",
"content": "$$130$$ "
}
],
[
{
"aoVal": "D",
"content": "$$600$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Common Factors and Common Multiples"
] | [
"We do prime factorization for $$25$$ and $$125$$ first. $$25={{5}^{2}}$$ and $$125={{5}^{3}}$$. We have three $$5$$'s, thus the least common multiple for $$25$$ and $$125$$ is $$125$$. We choose $$\\text{B}$$． "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 717 | 7233ba7cbb984793931332c2552a76a2 | [] | 1 | single_choice | Of the multiples of $$7$$ that exceed $$7$$, how many are factors of $$700$$? | [
[
{
"aoVal": "A",
"content": "$$99$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Theorem of the Number of Factors of a Number"
] | [
"Multiply $$7$$ by any of these: $$2$$, $$2^{2}$$, $$5$$, $$5^{2}$$, $$2\\times 5$$, $$2^{2}\\times 5$$, $$2\\times 5^{2}$$, or $$2^{2}\\times 5^{2}$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 723 | 9a54586b8cc94a149968bab4a6f05882 | [] | 0 | single_choice | Which of the following has an odd number of whole-number factors? | [
[
{
"aoVal": "A",
"content": "$$47$$ "
}
],
[
{
"aoVal": "B",
"content": "$$48$$ "
}
],
[
{
"aoVal": "C",
"content": "$$49$$ "
}
],
[
{
"aoVal": "D",
"content": "$$50$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers->Understanding Odd and Even Numbers"
] | [
"The whole-number factors of $$49$$ are $$1$$, $$7$$, and $$49$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 724 | e3dd6c2c4dea425d9a49122060ae6763 | [
"其它"
] | 1 | single_choice | There are three whole numbers $A$, $B$ and $C$. $A\times B=55$, $B\times C=100$. $A+B+C=$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$30$$ "
}
],
[
{
"aoVal": "B",
"content": "$$32$$ "
}
],
[
{
"aoVal": "C",
"content": "$$34$$ "
}
],
[
{
"aoVal": "D",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization"
] | [
"$55=5\\times 11$ $100=2\\times 2\\times 5\\times 5$ Because $B$ is the factor both number contains, $B=5$ Thus, $A=11$, $C=20$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 730 | 575386fd7f704fc48406725e383dbab6 | [] | 1 | single_choice | What is the difference between the largest single-digit prime number and the smallest three-digit prime number? | [
[
{
"aoVal": "A",
"content": "$$94$$ "
}
],
[
{
"aoVal": "B",
"content": "$$95$$ "
}
],
[
{
"aoVal": "C",
"content": "$$96$$ "
}
],
[
{
"aoVal": "D",
"content": "$$97$$ "
}
],
[
{
"aoVal": "E",
"content": "$$98$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization->Prime Factorization (Equations)"
] | [
"The largest single-digit prime number is $$7$$ and the smallest three-digit prime number is $$101$$; their difference is $$101 -7=94$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 731 | 64b7fd55ae7b4d23bca28562e1c24a98 | [] | 1 | single_choice | The product of \emph{all} the prime numbers less than $$10$$ is divisible by ． | [
[
{
"aoVal": "A",
"content": "$$70$$ "
}
],
[
{
"aoVal": "B",
"content": "$$60$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers->Determining Prime and Composite Numbers"
] | [
"The primes less than $$10$$ are $$2$$, $$3$$, $$5$$, and $$7$$. Their product is $$210$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 736 | 9a5e994657ea49db8570629c4e7df31c | [] | 1 | single_choice | The square root of $$49$$ is ． | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Perfect Square Numbers->Basic Applications of Square Numbers"
] | [
"The square root of $$49$$ is $$7$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 738 | a38e18aa86c9410da4b0f71cd07d74ab | [
"其它"
] | 1 | single_choice | Danni wants to use the numbers $1$, $3$, $7$, and $9$ to make prime numbers less than $100$. How many different prime numbers can she make? (She can use the same number more than once.) $$\textasciitilde$$ | [
[
{
"aoVal": "A",
"content": "$$8$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$14$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Prime and Composite Numbers"
] | [
"Using these $4$ numbers, we can make $20$ different numbers which are less than $100$. However, only these $12$ numbers are prime numbers. $3; 7; 11; 13; 17; 19; 31; 37; 71; 73; 79; 97$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 741 | 72544468182740249441b487e08a1120 | [] | 1 | single_choice | The numberis divisible by $$3\times3$$. | [
[
{
"aoVal": "A",
"content": "$$663$$ "
}
],
[
{
"aoVal": "B",
"content": "$$603$$ "
}
],
[
{
"aoVal": "C",
"content": "$$336$$ "
}
],
[
{
"aoVal": "D",
"content": "$$303$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"Add the digits to test: for $$603$$, $$6+0+3 =9$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 746 | ccdf978bf5bd4b4abc75beb07644ef7e | [
"其它"
] | 1 | single_choice | Ivan is puting 13 cakes into box A, box B and box C. It is known that there are odd number of cakes in both box A and box B. Could you tell whether the number of cakes in box C is even or odd number? | [
[
{
"aoVal": "A",
"content": "Odd "
}
],
[
{
"aoVal": "B",
"content": "Even "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Odd and Even Numbers"
] | [
"Number of cakes in box C: 13 (odd number) - odd - odd Number of odd character = 3 Therefore, the number of cakes in box C is odd. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 747 | dab0206dc920475bbed11b2c9c869187 | [] | 1 | single_choice | The ones digit of $$9\times 8\times 7\times 6\times 5\times 4\times 3\times 3\times 4\times 5\times 6\times 7\times 8\times 9$$ is． | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Numbers"
] | [
"Since $$5\\times 4 = 20$$, the ones digit of the given product must be $$0$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 748 | 695405d5736e4f419e7a5b933f9ff96b | [] | 1 | single_choice | The largest whole-number multiple of $$7$$ less than $$200$$ is ． | [
[
{
"aoVal": "A",
"content": "$$187$$ "
}
],
[
{
"aoVal": "B",
"content": "$$189$$ "
}
],
[
{
"aoVal": "C",
"content": "$$196$$ "
}
],
[
{
"aoVal": "D",
"content": "$$197$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Basic Concepts of Factors and Multiples"
] | [
"$$200\\div7\\textgreater28$$, so the largest such multiple is $$28\\times7 =196$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 749 | 4a4eb9d18d2d43f38d028d5417f88304 | [
"其它"
] | 2 | single_choice | The smallest number greater than $2$ that leaves a remainder of $2$ when divided by $3,4,5$, or $6$ lies between what numbers? (2012 AMC 8 Problem, Question \# 15) | [
[
{
"aoVal": "A",
"content": "$40$ and $50$ "
}
],
[
{
"aoVal": "B",
"content": "$51$ and $55$ "
}
],
[
{
"aoVal": "C",
"content": "$56$ and $60$ "
}
],
[
{
"aoVal": "D",
"content": "$61$ and $65$ "
}
],
[
{
"aoVal": "E",
"content": "$66$ and $99$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules"
] | [
"To find the answer to this problem, we need to find the least common multiple of $3,4,5,6$ and add $2$ to the result. The least common multiple of the four numbers is $60$ , and by adding $2$ , we find that that such number is $62$ . Now we need to find the only given range that contains $62$ . The only such range is answer (D), and so our final answer is $(\\text{D}) 61$ and $65$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 753 | 76e2528658994742bf63f02fc66e6c96 | [] | 1 | single_choice | What is the sum of all factors of $$24$$? | [
[
{
"aoVal": "A",
"content": "$$52$$ "
}
],
[
{
"aoVal": "B",
"content": "$$60$$ "
}
],
[
{
"aoVal": "C",
"content": "$$72$$ "
}
],
[
{
"aoVal": "D",
"content": "$$84$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Theorem of the Number of Factors of a Number"
] | [
"The sum of the factors is $$(3^{0}+3^{1})$$$$\\times (2^{0}+2^{1}+2^{2}+2^{3})=60$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 755 | 60671c37eb4149de9d4690e0daf2a887 | [] | 1 | single_choice | What is the greatest number of consecutive integers such that the sum of the digits of none of them is divisible by $$5$$? ． | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"Five consecutive numbers can be: $$12$$, $$13$$, $$14$$, $$15$$, and $$16$$ (without carrying) or $$17$$, $$18$$, $$19$$, $$20$$, $$21$$ (with carrying in the tens place). Without carrying, among each of the $$5$$ consecutive integers, we can find one whose sum of digits is a multiple of $$5$$. So we need to carry. To make the number of integers the greatest, we can start from a number whose remainder is $$1$$ when divided by five. And the most important thing is, after we write the fourth number, the carry appears. For example: $$56$$, $$57$$, $$58$$, $$59$$, $$60$$, $$61$$, $$62$$, $$63$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 758 | 578c303085a945e29296c5c87d9ce108 | [] | 1 | single_choice | The number $$1$$ million is less than the number． | [
[
{
"aoVal": "A",
"content": "$$1$$ thousand "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ thousand "
}
],
[
{
"aoVal": "C",
"content": "$$100$$ thousand "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ billion "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Numbers"
] | [
"The number $$1$$ billion is $$1$$ thousand times $$1$$ million. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 762 | d1846153d2f5412681d6acaa03506ea2 | [
"其它"
] | 2 | single_choice | $\overline{392AB}$ is a multiple of $45$, and $\overline{B34}$ is a three-digit even number. What is the sum of $A$ and $B$? | [
[
{
"aoVal": "A",
"content": "$$13$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$4$ or $13$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules"
] | [
"$45=5\\times9$ $\\overline{392AB}$ should be ended with $0$ or $5.$ But $0$ cannot be the first digit. Thus, $B$ can only be $5$ and $A$ should be $8.$ "
] | A |
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