Question 115
Suppose that $a$ and $b$ are positive integers such that $\operatorname{gcd}\left(a^{3}-b^{3},(a-b)^{3}\right)$ is not divisible by any perfect square except 1. Given that $1 \leq a-b \leq 50$, compute the number of possible values of $a-b$ across all such $a, b$.