<p>If \((27)^{999}\) is divided by 7, then the remainder is</p>
Step-by-Step Solution
Key Concept: Use Binomial Theorem by expressing 27 = 28 - 1 = 4(7) - 1, then expand (4·7 - 1)^999 to identify terms divisible by 7 and the remainder term.
<p><strong>Step 1:</strong> Express 27 in a form involving 7: <br/>27 = 28 - 1 = 4(7) - 1</p><p><strong>Step 2:</strong> Apply Binomial Theorem:<br/>(27)^999 = (4·7 - 1)^999 = Σ C(999,r)(4·7)^r(-1)^(999-r)</p><p><strong>Step 3:</strong> When divided by 7, all terms with r ≥ 1 contain factor 7 and are divisible by 7. Only the r = 0 term remains:<br/>C(999,0)(4·7)^0(-1)^999 = 1·1·(-1) = -1</p><p><strong>Step 4:</strong> Find remainder when -1 is divided by 7:<br/>-1 ≡ 6 (mod 7)</p><p><strong>Step 5:</strong> Verify: 27 ≡ -1 (mod 7), so (27)^999 ≡ (-1)^999 ≡ -1 ≡ 6 (mod 7)</p><p>∴ Answer: <strong>6</strong></p>
Correct Answer: A