Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Find the answer

For how many integers $a(1 \leq a \leq 200)$ is the number $a^{a}$ a square?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

107 If $a$ is even, we have $a^{a}=\left(a^{a / 2}\right)^{2}$ . If $a$ is odd, $a^{a}=\left(a^{(a-1) / 2}\right)^{2} \cdot a$ , which is a square precisely when $a$ is. Thus we have 100 even values of $a$ and 7 odd square values $\left(1^{2}, 3^{2}, \ldots, 13^{2}\right)$ for a total of 107.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.