Let $x=3^{a_1}5^{a_2}7^{a_3}$ and $y=3^{b_1}5^{b_2}7^{b_3}$ where $a_i,b_i$ are nonnegative integers such that $a_k + b_k \le 5$ for all $k$ find $$\sum \gcd(x,y).$$ where the sum is over all possible $x,y$.
Progress: I tried small cases on this problem but I didn't find a pattern. This is however equal to having $xy$ a divisor of $3^55^57^5$. I couldn't see what to do with this information, however. Maybe there is a systematic way to do count this? I also tried multiplying geometric series $$(1+3+9+\cdots)(1+5+25+\cdots)(1+7+49\cdots)$$ for the desired term, but that didn't work either.
HINT
It's the product of three terms of the form $$\sum_{a=0}^5\sum_{b=0}^{5-a}p^{\min(\{a,b\})}$$
for $p=3,5,7$, so you just need to figure out how to compute those.