<p>For all \(n \in \mathbb{N}\), \(3 \cdot 5^{2n+1} + 2^{3n+1}\) is divisible by __________.</p>
Step-by-Step Solution
Key Concept: Use mathematical induction or modular arithmetic to find a common divisor. Test small values of n to identify the pattern, then verify divisibility by checking the expression modulo potential divisors.
<p><strong>Step 1:</strong> Test small values of n to identify the divisor.</p><p>For n = 0: 3·5¹ + 2¹ = 15 + 2 = 17</p><p>For n = 1: 3·5³ + 2⁴ = 3·125 + 16 = 375 + 16 = 391 = 17 × 23</p><p>For n = 2: 3·5⁵ + 2⁷ = 3·3125 + 128 = 9375 + 128 = 9503 = 17 × 559</p><p><strong>Step 2:</strong> Verify using mathematical induction that 3·5^(2n+1) + 2^(3n+1) ≡ 0 (mod 17).</p><p><strong>Base case (n=0):</strong> 3·5 + 2 = 17 ≡ 0 (mod 17) ✓</p><p><strong>Inductive step:</strong> Assume 3·5^(2k+1) + 2^(3k+1) ≡ 0 (mod 17) for n = k.</p><p>For n = k+1: 3·5^(2k+3) + 2^(3k+4) = 25·3·5^(2k+1) + 8·2^(3k+1)</p><p>Since 25 ≡ 8 (mod 17), both terms can be rewritten and shown to be ≡ 0 (mod 17).</p><p>∴ Answer: 17</p>
Correct Answer: 17