Question

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

• A. $4$
• B. $8$
• C. $14$
• D. $6$

Answer 1

The correct option is B $8$

$\frac{2^{403}}{15}=2^3\times \left(\frac{2^{400}}{15}\right)=8\times \left(\frac{{16}^{100}}{15}\right)$
$=\frac{8}{15}(1+15{)}^{100}$
Using binomial theorem we can write above expression as
$\frac{8}{15}(1+15.n)$, $n\in N$
$=\frac{8}{15}+8.n$
So the fractional part is $\frac{8}{15}=\frac{k}{15}$
Hence, the value of $k=8$