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Stage 4 · Number theory

10 problems · AMC 12 late, AIME early · mathsolympiadprep.com

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

1. Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72$.

   HMMT November 2024

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2. How many two-digit prime numbers have the property that both digits are also primes?

   HMMT November 2014

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3. How many multiples of $7$ between $10^{6}$ and $10^{9}$ are perfect squares?

   United States competition problems

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4. Find the smallest positive integer that is twice a perfect square and three times a perfect cube.

   Annual Harvard-MIT Mathematics Tournament

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5. Find the number of positive divisors $d$ of $15! = 15 \cdot 14 \cdots 2 \cdot 1$ such that $\operatorname{gcd}(d, 60) = 5$.

   HMMT 2013

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6. Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ (that is, $1$, powers of $2$, and powers of $2017$ are thus contained in $S$). Compute
   $$\sum_{s \in S} \frac{1}{s}.$$

   HMMT November

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7. Find the smallest $n$ such that $n!$ ends with 10 zeroes.

   HMMT November

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8. For positive integers $m, n$, let $\operatorname{gcd}(m, n)$ denote the largest positive integer that is a factor of both $m$ and $n$. Compute
   $$\sum_{n=1}^{91} \operatorname{gcd}(n, 91) .$$

   Harvard-MIT November Tournament

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9. Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11.

   HMMT November

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10. What is the remainder when $3^{2020}$ is divided by $73$?

    Philippine Mathematical Olympiad

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Answer key — Stage 4 · Number theory

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

1. $96$ open
2. $4$ open
3. $4375$ open
4. $648$ open
5. $36$ open
6. $2017/1008$ open
7. $45$ open
8. $325$ open
9. $72$ open
10. $8$ 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.

CC-BY-4.0 · Apache-2.0