Factoring Combinations

The number of unique positive integers $m$ will correspond to the number of positive factor pairs of $16 \cdot 225 = 3600$ . Since $3600 = 2^4 \cdot 3^2 \cdot 5^2$ , it has $(4 + 1)(2 + 1)(2 + 1) = 45$ factors, and $23$ factor pairs. There are therefore $\boxed{23}$ unique integer solutions for $m$ .