When 10^100 is divided by 7, the remainder is

Options

1. 3
2. 6
3. 1
4. 4

Correct Answer

4

Explanation

By Fermat's Little Theorem, since 7 is prime and gcd(10,7)=1, we have 10^6 ≡ 1 (mod 7). Now 100 = 16×6 + 4, so 10^100 = (10^6)^16 × 10^4 ≡ 1^16 × 10^4 ≡ 10^4 (mod 7). Computing: 10 ≡ 3 (mod 7), 10² ≡ 9 ≡ 2 (mod 7), 10^4 ≡ 2² ≡ 4 (mod 7). Remainder is 4.

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