jmc

We start by writing out some powers of five modulo 7. $$\begin{array}{rl}{5}^1& \equiv 5(\mathrm{mod}7)\\ 5^2& \equiv 4(\mathrm{mod}7)\\ 5^3& \equiv 6(\mathrm{mod}7)\\ 5^4& \equiv 2(\mathrm{mod}7)\\ 5^5& \equiv 3(\mathrm{mod}7)\\ 5^6& \equiv 1(\mathrm{mod}7)\end{array}$$Therefore, we have that $5^6\equiv 1$ modulo 7. Thus, $5^{2010}\equiv (5^6{)}^{335}\equiv 1^{335}\equiv \boxed{1}$ modulo 7.