When $10^{100}$ is divided by 7, the remainder is:

10 leaves remainder 3 on division by 7, so $10^{100}$ ≡ $3^{100}$ (mod 7). Now $3^6=729=7\times 104+1$, so the powers of 3 repeat every 6 steps. As $100=6\times 16+4$, we need $3^4=81=7\times 11+4$. The remainder is 4.