jmc

If we look at the terms of the sequence mod 4, we see that they follow a pattern of period 6: $$\begin{array}{rl}{F}_1& \equiv 1(\mathrm{mod}4),\\ F_2& \equiv 1(\mathrm{mod}4),\\ F_3& \equiv 2(\mathrm{mod}4),\\ F_4& \equiv 3(\mathrm{mod}4),\\ F_5& \equiv 1(\mathrm{mod}4),\\ F_6& \equiv 0(\mathrm{mod}4),\\ F_7& \equiv 1(\mathrm{mod}4),\\ F_8& \equiv 1(\mathrm{mod}4),\dots \end{array}$$ Then we see that the terms repeat. Therefore, the $100^{\text{th}}$ term is the same as the $4^{\text{th}}$ term, and thus has a remainder of $\boxed{3}$ when divided by 4.