1) Prove that if n is a square, then each exponent in its prime-power decomposition is even.
and
2) Prove that if each exponent in the prime-power decomposition of n is even, then n is square.
I do the first one and leave the second one for thee to think about.
(Assuming )
If is a square it means . Now and so it can be written as by prime decomposition. This means, . And so exponents in prime decomposition of are even.
Similar problem: Once you prove the above problem try proving a stronger result. That is an -th power if and only if each prime in the decomposition of is a multiple of .