<p>The fractional part of \(2^{4n}/15\) is (\(n \in \mathbb{N}\))</p>
Step-by-Step Solution
Key Concept: Use binomial expansion to write 2^(4n) = 16^n = (15+1)^n, then apply binomial theorem to identify the remainder when divided by 15. The fractional part equals the remainder divided by 15.
<p><strong>Step 1:</strong> Express 2^(4n) using binomial expansion.</p><p>2^(4n) = 16^n = (15+1)^n</p><p><strong>Step 2:</strong> Apply binomial theorem: (15+1)^n = C(n,0)·15^n + C(n,1)·15^(n-1) + ... + C(n,n-1)·15 + C(n,n)</p><p><strong>Step 3:</strong> Observe that all terms except the last are divisible by 15:</p><p>(15+1)^n = 15k + 1, where k is some positive integer</p><p><strong>Step 4:</strong> Therefore: 2^(4n) = 15k + 1</p><p>The remainder when 2^(4n) is divided by 15 is always 1.</p><p><strong>Step 5:</strong> The fractional part of 2^(4n)/15 = 1/15</p><p>∴ Answer: A (which is 1/15)</p>
Correct Answer: A