HMMT 2014 U21

Compute the number of ordered quintuples of nonnegative integers ( $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ ) such that $0 \leq a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \leq 7$ and 5 divides $2^{a_{1}}+2^{a_{2}}+2^{a_{3}}+2^{a_{4}}+2^{a_{5}}$.