Question

The fractional part of $\frac{2^{4n}}{15}$ is $(n\in N)$

• A. $\frac{1}{15}$
• B. $\frac{2}{15}$
• C. $\frac{4}{15}$
• D. none of these

Answer 1

Answer: (A)

$\frac{1}{15}$

The fractional part of $\frac{2^{4n}}{15}$ basically asks us to find out the remainder when $2^{4n}$ is divided by $15.$

$\therefore 2^{4n}={(2}^4){}^n={16}^n={(15+1)}^n$

$\Rightarrow {(15+1)}^n=\left\{{15}^n{+}^n{C}_r{15}^{n-1}+...{.}^n{C}_{n-1}15{+}^n{C}_n.1\right\}$

$=15\lambda +1;$ where $\lambda \in I$

$\therefore$ On dividing ${16}^n$ by $15$, we will get $\lambda$ as quotient $\&$ $1$ as remainder.

Hence, fractional part will always be $\frac{1}{15}$

Hence, the answer is $\frac{1}{15}.$