<p>Find the remainder when \(34562^{222}\) is divided by 7.</p>
Step-by-Step Solution
Key Concept: Use Fermat's Little Theorem: for prime p and gcd(a,p)=1, we have a^(p-1) ≡ 1 (mod p). Since 6 divides 222, reduce 34562 modulo 7 first, then apply the theorem to simplify the exponent.
<p><strong>Step 1:</strong> Reduce the base modulo 7: 34562 = 7×4937 + 3, so 34562 ≡ 3 (mod 7)</p><p><strong>Step 2:</strong> We need to find 3^222 (mod 7). By Fermat's Little Theorem, since 7 is prime and gcd(3,7) = 1: 3^6 ≡ 1 (mod 7)</p><p><strong>Step 3:</strong> Express the exponent in terms of 6: 222 = 6×37 + 0, so 222 is divisible by 6</p><p><strong>Step 4:</strong> Therefore, 3^222 = (3^6)^37 ≡ 1^37 ≡ 1 (mod 7)</p><p>∴ Answer: 1</p>
Correct Answer: 4