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