Question

$2^{3n} -7n -1$ is divisible by

• A. $64$
• B. $36$
• C. $49$
• D. $25$

Answer 1

Answer: (C)

Let $P\left(n\right) = 2^{3n} - 7^{n} - 1 $
$\therefore P\left(1\right) = 0, P\left(2\right) = 49$
$P\left(1\right)$ and $P\left(2\right)$ are divisible by $49$.
Let $P\left(k\right) \equiv 2^{3k}-7k-1= 49\lambda\quad...\left(i\right)$
$\therefore P\left(k+ 1\right) = 23^{3k+ 3}-7\left(k+ 1\right)- 1 $
$= 8\left(49\lambda + 7k+ 1\right)-7k-8\quad$ [using $\left(i\right)$]
$ = 49\left(8\lambda\right) + 49k = 49\left(8\lambda + k\right) $
Hence, by the principle of mathematical induction
$ 2^{3n} - 7n - 1$ is divisible by $49$.