<p>The remainder left out when \(8^{2n} - (62)^{2n+1}\) is divided by 9 is</p>
Step-by-Step Solution
Key Concept: Use modular arithmetic with Fermat's Little Theorem or Euler's theorem. Since φ(9) = 6, we have a^6 ≡ 1 (mod 9) for gcd(a,9)=1. Reduce 8 and 62 modulo 9 first, then find the pattern of powers.
<p><strong>Step 1:</strong> Reduce bases modulo 9.</p><p>8 ≡ -1 (mod 9) and 62 ≡ -1 (mod 9)</p><p><strong>Step 2:</strong> Substitute into the expression.</p><p>8^(2n) - 62^(2n+1) ≡ (-1)^(2n) - (-1)^(2n+1) (mod 9)</p><p><strong>Step 3:</strong> Evaluate powers of -1.</p><p>(-1)^(2n) = 1 (always, since 2n is even)</p><p>(-1)^(2n+1) = -1 (always, since 2n+1 is odd)</p><p><strong>Step 4:</strong> Compute the result.</p><p>1 - (-1) = 1 + 1 = 2 (mod 9)</p><p>∴ Answer: 2</p>
Correct Answer: C