South African Mathematics Olympiad

The last digits of the first few powers of $2$ are $2$, $4$, $8$, $6$, $2$, $4$, ..., so the digits repeat in cycles of length $4$. Since the remainder after dividing $2015$ by $4$ is $3$, it follows that the last digit of $2^{2015}$ is the same as the last digit of $2^3$, which is $8$. Finally, the last digit of any power of $5$ is $5$, so the last digit of $2^{2015}+5^{2015}$ is equal to the last digit of $8+5$, which is $3$.