Ok, so (2^3)(5^2) = 200, which is 2*100. The 100 is a perfect square so only the 2 has to be a factor of m.
The only way to make the 13 a factor of a perfect square is to multiply it by itself (because it's a prime number). So 13 has to be another factor of m. Then we look at the 7^9 which has an odd power. You need an EVEN power to form a perfect square, so just multiply it by another 7 to get 7^10. That means 7 is another factor of m.