First slide

Binomial theorem for positive integral Index

Question

The fractional part of $\frac{2^{4 n}}{15}$ is

Moderate

A

$\frac{2}{15}$
B

$\frac{1}{15}$
C

$\frac{4}{15}$
D

none of these

Solution

We have, $\frac{2^{4 n}}{15} = \frac{16^{n}}{15} = \frac{(1 + 15)^{n}}{15}$
$\begin{array}{l} = \frac{1 +^{n} C_{1} 15 +^{n} C_{2} 15^{2} + \dots +^{n} C_{n} 15^{n}}{15} \\ = \frac{1 + 15 k}{15}, where k \in N \\ = \frac{1}{15} + k \end{array}$
$∴ Fractional part of \frac{2^{4 n}}{15} is \frac{1}{15}$

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