First slide

Introduction to P.M.I

Question

For all $n \in N, 3^{3 n} - 26^{n} - 1$ is divisible by

Easy

A

24
B

64
C

17
D

676

Solution

We have,
$\begin{array}{l} 3^{3 n} - 26 n - 1 = 27^{n} - 26 n - 1 \\ \Rightarrow 3^{3 n} - 26 n - 1 = (1 + 26)^{n} - 26 n - 1 \\ \Rightarrow 3^{3 n} - 26 n - 1 =^{n} C_{2} \times 26^{2} +^{n} C_{3} \times 26^{3} + \dots +^{n} C_{n} \times 26^{n} \end{array}$
Clearly, RHS is divisible 26² i.e. 676.

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