<p>\(3^{51}\) when divided by 8 leaves the remainder</p>
Step-by-Step Solution
Key Concept: Use the Binomial Theorem to express 3^51 as (1+2)^51, then expand and find the remainder when divided by 8 by identifying which terms are divisible by 8.
<p><strong>Step 1:</strong> Express 3^51 using Binomial Theorem</p><p>3^51 = (1+2)^51 = Σ C(51,r)·1^(51-r)·2^r = Σ C(51,r)·2^r</p><p><strong>Step 2:</strong> Identify terms modulo 8</p><p>For r ≥ 3: 2^r is divisible by 8, so C(51,r)·2^r ≡ 0 (mod 8)</p><p><strong>Step 3:</strong> Extract relevant terms</p><p>3^51 ≡ C(51,0)·2^0 + C(51,1)·2^1 + C(51,2)·2^2 (mod 8)</p><p>= 1·1 + 51·2 + (51·50/2)·4 (mod 8)</p><p>= 1 + 102 + 5100 (mod 8)</p><p><strong>Step 4:</strong> Simplify modulo 8</p><p>102 = 96 + 6 ≡ 6 (mod 8)</p><p>5100 = 5096 + 4 ≡ 4 (mod 8)</p><p>3^51 ≡ 1 + 6 + 4 = 11 ≡ 3 (mod 8)</p><p>∴ Answer: 3</p>
Correct Answer: D