Let $F_k(a,b) = (a+b)^k - a^k - b^k$ and let $S = \{1,2,3,4,5,6,7,8,9,10\}$. For how many ordered pairs $(a,b)$ with $a,b \in S$ and $a \leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?

Let's think step by step and output the final answer within \boxed{}.
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