Competition problem — Number theory

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Number theory Difficulty 5.1 AIME, harder Find the answer

Let a positive integer $n$ be called a cubic square if there exist positive integers $a, b$ with $n=\operatorname{gcd}\left(a^{2}, b^{3}\right)$ . Count the number of cubic squares between 1 and 100 inclusive.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

This is easily equivalent to $v_{p}(n) \not \equiv 1,5(\bmod 6)$ for all primes $p$ . We just count: $p \geq 11 \Longrightarrow v_{p}(n)=1$ is clear, so we only look at the prime factorizations with primes from $\{2,3,5,7\}$ . This is easy to compute: we obtain 13.

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