Question

For all $n\in N,2^{4n}-15n-1$ is divisibl;e by

• A. $225$
• B. $125$
• C. $325$
• D. None of these

Answer 1

Answer: (A)

$225$

We have $2^{4n}={\left(2^4\right)}^n={\left(16\right)}^n={(1+15)}^n$

$\therefore 2^{4n}=1{+}^n{C}_1\times 15{+}^n{C}_2\times {15}^2{+}^n{C}_3\times {15}^3+...\Rightarrow 2^{4n}-1-15n={15}^2[{}^n{C}_2{+}^n{C}_3\times 15+...]$

$=225K,$ where $K$ is an integer.

Hence $2^{4n}-1-15n$ is divisible by $225$.