<p>The remainder left out when \(8^{2n} - (62)^{2n+1}\) is divided by 9 is</p>
Step-by-Step Solution
Key Concept: Use binomial theorem to expand (62)^(2n+1) in terms of modulo 9, recognizing that 62 ≡ -1 (mod 9) and 8 ≡ -1 (mod 9), then evaluate the resulting expression.
<p><strong>Step 1:</strong> Express both terms modulo 9.</p><p>8 ≡ -1 (mod 9) and 62 = 63 - 1 = 7(9) - 1 ≡ -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 the powers.</p><p>(-1)^(2n) = 1 (since 2n is even)</p><p>(-1)^(2n+1) = -1 (since 2n+1 is odd)</p><p><strong>Step 4:</strong> Calculate the remainder.</p><p>1 - (-1) = 1 + 1 = 2</p><p>∴ The remainder when 8^(2n) - (62)^(2n+1) is divided by 9 is <strong>2</strong></p>
Correct Answer: A