If the fractional part of the number

$\frac{2^{403}}{15}=\frac{2^{3} \cdot\left(2^{4}\right)^{100}}{15}=\frac{8}{15}(15+1)^{100}$

$=\frac{8}{15}(15 \lambda+1)=8 \lambda+\frac{8}{15}$

$\because 8 \lambda$ is integer

$\Rightarrow$ fractional part of $\frac{2^{403}}{15}$ is $\frac{8}{15} \Rightarrow k=8$