What are the number of integers a $1 \le a \le 100$ such that $a^a$ is a perfect square.

Question

What are the number of integers a such that $1 \le a \le 100$ and $a^a$ is a perfect square.

I think the answer should be 51 since a can be 1 and then 2,4,6,...100.

Is the answer correct?

Answer 1

No, it's not correct: If $a$ is an odd perfect square, then so is $a^a$. For example,

$$9^9 = (3^2)^9 = (3^9)^2$$

Answer 2

The even numbers - 50 of them

The odd numbers, which are perfect squares: 1, 9, 25, 49, 81 - another.

Altogether 55