Three-digit powers of $2$ and $5$ are used in this "cross-number" puzzle. The number for '1. Down' is a three-digit power of $5$, and the number for '2. Across' is a three-digit power of $2$. What is the only possible digit for the outlined square, which is the intersection of the two numbers?
for the puzzle grid and 072e6a8e-c47f-42d0-9c79-882af0e968e2_1.png for the clues.