Competition problem — Number theory

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Number theory Difficulty 4.3 AIME Find the answer United States

Problem:

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

Solution:

If $a$ is even, we have $a^{a} = \left(a^{a / 2}\right)^{2}$ , which is always a square.

If $a$ is odd, $a^{a} = \left(a^{(a-1) / 2}\right)^{2} \cdot a$ , which is a square precisely when $a$ is a square.

Thus we have 100 even values of $a$ and 7 odd square values ( $1^{2}, 3^{2}, \ldots, 13^{2}$ ) for a total of 107.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.