Basic Mathematics & Logarithm
Mathematical Induction - Divisibility
Grade 11
Question:
<p>For all \(n \in \mathbb{N}\), \(3^{3n} - 26n - 1\) is divisible by</p>
<p>(a) 24</p>
<p>(b) 64</p>
<p>(c) 17</p>
<p>(d) 676</p>
Step-by-Step Solution
Key Concept: Use the binomial theorem to expand $(1 + 26)^n$ and show that every term except the linear term is divisible by $26^2 = 676$.
<p><strong>Solution:</strong></p><p>We have, $3^{3n} - 26n - 1 = 27^n - 26n - 1$</p><p>$\Rightarrow 3^{3n} - 26n - 1 = (1 + 26)^n - 26n - 1$</p><p>$\Rightarrow 3^{3n} - 26n - 1 = \binom{n}{2} \cdot 26^2 + \binom{n}{3} \cdot 26^3 + \ldots + \binom{n}{n} \cdot 26^n$</p><p>Clearly, RHS is divisible by $26^2$ i.e., 676.</p><p>∴ Answer is (d).</p>
Correct Answer: D