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Stage 3 · Number theory

10 problems · AMC 10/12, early questions · mathsolympiadprep.com

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

1. For how many integers $a$ with $1 \leq a \leq 10$ is $a^{2014}+a^{2015}$ divisible by 5?

   CEMC Fermat

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2. For how many pairs $(m, n)$ with $m$ and $n$ integers satisfying $1 \leq m \leq 100$ and $101 \leq n \leq 205$ is $3^{m}+7^{n}$ divisible by 10?

   CEMC Cayley

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3. For each positive integer $n$, define $S(n)$ to be the smallest positive integer divisible by each of the positive integers $1, 2, 3, \ldots, n$. How many positive integers $n$ with $1 \leq n \leq 100$ have $S(n) = S(n+4)$?

   CEMC Cayley

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4. The integer $N$ is the smallest positive integer that is a multiple of 2024, has more than 100 positive divisors (including 1 and $N$), and has fewer than 110 positive divisors (including 1 and $N$). What is the sum of the digits of $N$?

   CEMC Fermat

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5. If $m$ and $n$ are positive integers with $n > 1$ such that $m^{n} = 2^{25} \times 3^{40}$, what is $m + n$?

   CEMC Fermat

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6. For how many odd integers $k$ between 0 and 100 does the equation $2^{4m^{2}}+2^{m^{2}-n^{2}+4}=2^{k+4}+2^{3m^{2}+n^{2}+k}$ have exactly two pairs of positive integers $(m, n)$ that are solutions?

   CEMC Fermat

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7. There are $F$ fractions $\frac{m}{n}$ with the properties: $m$ and $n$ are positive integers with $m<n$, $\frac{m}{n}$ is in lowest terms, $n$ is not divisible by the square of any integer larger than 1, and the shortest sequence of consecutive digits that repeats consecutively and indefinitely in the decimal equivalent of $\frac{m}{n}$ has length 6. We define $G=F+p$, where the integer $F$ has $p$ digits. What is the sum of the squares of the digits of $G$?

   CEMC Fermat

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8. Teddy has $10$ rectangular
   blocks each of which measures $\text{3 cm}$ by $\text{4 cm}$ by
   $\text{5 cm}$. He builds a stack
   that is exactly $\text{21 cm}$ high,
   where each block can be stacked on any of its faces. What is the
   smallest number of blocks that Teddy could use to make this stack?

   1. A$4$
   2. B$5$
   3. C$6$
   4. D$7$
   5. E$8$

   CEMC Cayley 2025

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9. Find the remainder when $19^{17^{15}}$ is divided by 100.

   Hong Kong Preliminary Selection Contest

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10. The remainder when 111 is divided by 10 is 1. The remainder when
    111 is divided by the positive integer $n$ is 6. The number of possible values of
    $n$ is

    1. A$5$
    2. B$8$
    3. C$7$
    4. D$6$
    5. E$4$

    CEMC Fermat 2022

    Number theory Solution and answer checking →

Answer key — Stage 3 · Number theory

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

1. $4$ open
2. $2625$ open
3. $11$ open
4. $27$ open
5. $209957$ open
6. $18$ open
7. $181$ open
8. $B$ open
9. $59$ open
10. $A$ open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.

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