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Stage 2 · Number theory

10 problems · · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

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$91$

   CEMC Fermat 2025

   Number theory Solution and answer checking →

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$6$

   CEMC Pascal 2014

   Number theory Solution and answer checking →

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?

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

   CEMC Fermat 2021

   Number theory Solution and answer checking →

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$36$

   CEMC Cayley 2022

   Number theory Solution and answer checking →

5. The product $8 \times 48 \times 81$ is divisible by $6^k$. The largest possible integer value of $k$ is

   1. A$7$
   2. B$6$
   3. C$5$
   4. D$4$
   5. E$3$

   CEMC Fermat 2019

   Number theory Solution and answer checking →

6. 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$90 \text{ cm}^3$

   CEMC Cayley 2026

   Number theory Solution and answer checking →

7. 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 $N$?

   CEMC Pascal

   Number theory Solution and answer checking →

8. 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$22$

   CEMC Gauss (Grade 7) 2020

   Number theory Solution and answer checking →

9. 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$7$

   CEMC Fermat 2019

   Number theory Solution and answer checking →

10. 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$5$

    CEMC Pascal 2021

    Number theory Solution and answer checking →

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. $C$ open
6. $A$ open
7. $19$ open
8. $D$ open
9. $B$ open
10. $C$ open

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