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Stage 2 · Number theory

10 problems · · mathsolympiadprep.com

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1. The current calendar year, $2025$, is a perfect square. In $n$ years from 2025, the calendar year will
   again be a perfect square. The smallest possible value of $n$ is

   1. A$2025$
   2. B$100$
   3. C$9$
   4. D$46$
   5. E$91$

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2. If $2 \times 2 \times 3 \times 3 \times 5 \times 6 = 5\times 6 \times n \times n$, then $n$ could equal

   1. A$2$
   2. B$3$
   3. C$4$
   4. D$5$
   5. E$6$

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3. The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer?

   $5$
   $7$
   $6$
   $8$
   $9$

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4. A rectangle has positive integer side lengths and an area of 24.
   The perimeter of the rectangle cannot be

   1. A$20$
   2. B$22$
   3. C$28$
   4. D$50$
   5. E$36$

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5. The edge lengths of a rectangular prism are integers. Three of
   its faces have areas $12 $\text{}$
   cm}^2$,$20 $\text{}$ cm}^2$ and
   $15 \text{ cm}^2$. The volume of the
   prism is

   1. A$60 \text{ cm}^3$
   2. B$30 \text{ cm}^3$
   3. C$3600 \text{ cm}^3$
   4. D$120 \text{ cm}^3$
   5. E$90 \text{ cm}^3$

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6. When the three-digit positive integer $N$ is divided by 10, 11, or 12, the remainder is 7. What is the sum of the digits of $N$?

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7. The positive divisors of 12 (other than itself) are $1, 2, 3, 4$, and 6. Their sum, $1+2+3+4+6$, is greater than 12. An abundant number is a number for which the sum of its positive divisors (other than itself) is greater than the number itself. This means that 12 is an abundant number. Which of the following is also an abundant number?

   1. A$8$
   2. B$10$
   3. C$14$
   4. D$18$
   5. E$22$

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8. The digits $2$, $3$, $5$, $7$, and $8$ can be used, each exactly once, to form many five-digit integers. Of these integers, $N$ is the one that is as close as possible to 30 000. What is the tens digit of $N$?

   1. A$2$
   2. B$5$
   3. C$3$
   4. D$8$
   5. E$7$

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9. How many integers between 100 and 300 are multiples of both 5 and 7, but are not multiples of 10?

   1. A$1$
   2. B$2$
   3. C$3$
   4. D$4$
   5. E$5$

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10. If $\$2^{11}\times 6^5=4^x\times$
    3^y for some positive integers x$and$y$, then the value of$x+y$ is

    1. A$10$
    2. B$11$
    3. C$12$
    4. D$13$
    5. E$14$

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Answer key — Stage 2 · Number theory

Worked solutions for every problem are on the site, one page per problem.

1. $E$ open
2. $E$ open
3. $C$ open
4. $E$ open
5. $A$ open
6. $19$ open
7. $D$ open
8. $B$ open
9. $C$ open
10. $D$ open

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