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  4. If the fractional part of the number (2403/15) is (k/15), then k is equal to :

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

JEE MainJEE Main 2019Binomial Theorem

Solution:

$\frac{2^{403}}{15} = \frac{2^{3}. \left(2^{4}\right)^{100}}{15} = \frac{8}{15} \left(15+1\right)^{100} $
$ = \frac{8}{15} \left(15 \lambda +1\right) = 8 \lambda + \frac{8}{15} $
$ \because 8 \lambda$ is integer
$ \Rightarrow $ fractional part of $ \frac{2^{403}}{15} \frac{8}{15} \Rightarrow k = 8 $