As perfect as a square! 1

A perfect square is the product of some integer with itself.

Case 1: When $n = 2k$ for some positive integer $k$

We know that

$(5n)^{5n} = (10k)^{10k} = ((10k)^{5k})^{2}$ is a perfect square. Therefore we know that there are 100 even numbers and they all satisfy the property that $(5n)^{5n}$ is a perfect square.

Case 2: When $n = 5k^{2}$ for some positive integer $k$

Similarly,

$(5n)^{5n} = ((5k)^{25k^{2}})^{2}$ is a perfect square. Therefore the values of $k$ such that $1 ≤ n ≤ 200$ are $k = 1, 2, ... , 6$ .

However when $k = 2, 4, 6, n$ is an even number, but we have already counted even numbers.

Thus there are $100 + 3 = 103$ such numbers.