<p>The remainder when the number \(3^{256} - 3^{12}\) is divided by 8 is</p>
Step-by-Step Solution
Key Concept: Use the Binomial Theorem to express powers of 3 in the form (1+2)^n, then find remainders modulo 8. Alternatively, find the pattern of powers of 3 modulo 8 using periodicity.
<p><strong>Step 1:</strong> Identify the pattern of powers of 3 modulo 8.</p><p>3¹ ≡ 3 (mod 8)</p><p>3² = 9 ≡ 1 (mod 8)</p><p>3³ = 27 ≡ 3 (mod 8)</p><p>3⁴ = 81 ≡ 1 (mod 8)</p><p>Pattern: 3^(2k) ≡ 1 (mod 8) and 3^(2k+1) ≡ 3 (mod 8)</p><p><strong>Step 2:</strong> Apply the pattern to our exponents.</p><p>256 is even: 3^256 ≡ 1 (mod 8)</p><p>12 is even: 3^12 ≡ 1 (mod 8)</p><p><strong>Step 3:</strong> Calculate the remainder.</p><p>3^256 - 3^12 ≡ 1 - 1 ≡ 0 (mod 8)</p><p><strong>Verification using Binomial Theorem:</strong> 3² = (1+2)² = 1 + 4 + 4 ≡ 1 (mod 8). Since both exponents are even, 3^256 - 3^12 ≡ 0 (mod 8).</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: A