Problem 497

The units digits of powers of 3 cycle $3,9,7,1$ and the units digits of powers of 7 cycle $7,9,3,1$. For $3^{m}+7^{n}$ to be divisible by 10, one of the following must be true: units digit of $3^{m}$ is 3 and $7^{n}$ is 7, or 9 and 1, or 7 and 3, or 1 and 9. The number of possible pairs $(m, n)$ is $27 \times 25+26 \times 25+26 \times 25+26 \times 25=2625$.