Question

For all $n\in N,7^{2n}-48n-1$ is divisible by

• A. $2304$
• B. $576$
• C. $24$
• D. $18$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $2304$
B $576$
C $24$
D $18$

$a=7^{2n}-48n-1=-1-48n+{(1+48)}^n$
$\Rightarrow a=-1-48n+(1{+}^n{C}_{1}48{+}^n{C}_2{48}^2+...{+}^n{C}_n{48}^n)={48}^2({}^n{C}_2{+}^n{C}_{3}48+...{+}^n{C}_n{48}^{n-2})$
$\Rightarrow a=2304({}^n{C}_2{+}^n{C}_{3}48+...{+}^n{C}_n{48}^{n-2})$
Therefore, $a$ is divisible by $2304$.
Ans: A