TITLE: What are the number of integers a $1 \le a \le 100$ such that $a^a$ is a perfect square.
QUESTION [6 upvotes]: 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?

REPLY [10 votes]: No, it's not correct: If $a$ is an odd perfect square, then so is $a^a$. For example,
$$9^9 = (3^2)^9 = (3^9)^2$$