jmc

We begin by computing the remainder of some small powers of $7$. As $7^0=1,7^1=7,$ and $7^2=49$, then $7^3=49\cdot 7=343$ leaves a remainder of $43$ after division by $100$, and $7^4$ leaves the remainder that $43\cdot 7=301$ does after division by $100$, namely $1$. The sequence of powers thus repeat modulo $100$ again. In particular, the remainder that the powers of $7$ leave after division by $100$ is periodic with period $4$. Then, $7^{2010}=7^{4\cdot 502+2}$ leaves the same remainder as $7^2$ does after division by $100$, which is $\boxed{49}$.