jmc

A positive integer is a factor of $2^4\cdot 7^9$ if and only if its prime factorization is of the form $2^a\cdot 7^b$ for exponents $a$ and $b$ satisfying $0\le a\le 4$ and $0\le b\le 9$. A positive integer is a perfect square if and only if the exponents in its prime factorization are even. A positive integer is even if and only if the exponent of 2 in its prime factorization is at least 1. Therefore, we may choose $a=2$ or $4$ and $b=0,$ $2,$ $4,$ $6,$ or $8.$ Since we have 2 choices for $a$ and 5 choices for $b$, there are $2\times 5=\boxed{10}$ ways to make these two decisions.