The remainder when $7^{100}$ is divided by 5 is:

7 leaves remainder 2 on division by 5, so $7^{100}$ ≡ $2^{100}$ (mod 5). Powers of 2 repeat with remainders 2, 4, 3, 1 in a cycle of 4. Since 100 is a multiple of 4, the remainder is 1.