If the fractional part of the number
Question:

If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{\mathrm{k}}{15}$, then $\mathrm{k}$ is equal to:

1. (1) 6

2. (2) 8

3. (3) 4

4. (4) 14

Correct Option: , 2

Solution:

$2^{403}=2^{400} \cdot 2^{3}$

$=2^{4 \times 100} \cdot 2^{3}$

$=\left(2^{4}\right)^{100} \cdot 8$

$=8\left(2^{4}\right)^{100}=8(16)^{100}$

$=8(1+15)^{100}$

$=8+15 \mu$

When $2^{403}$ is divided by 15 , then remainder is $8 .$

Hence, fractional part of the number is $\frac{8}{15}$

Therefore value of $k$ is 8