The fractional part of the number frac{4^{202 | vedclass

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Question

(B) We need to find the fractional part of $\frac{4^{2022}}{15}$.Note that $4^{2022} = (4^2)^{1011} = 16^{1011}$.We can write $16$ as $(15 + 1)$.So,$1

Vedclass · Accepted Answer

$\frac{1}{15}$

Vedclass · Answer

(B) We need to find the fractional part of $\frac{4^{2022}}{15}$.Note that $4^{2022} = (4^2)^{1011} = 16^{1011}$.We can write $16$ as $(15 + 1)$.So,$16^{1011} = (15 + 1)^{1011}$.Using the Binomial Theorem,$(15 + 1)^{1011} = \binom{1011}{0} 15^{1011} + \binom{1011}{1} 15^{1010} + \dots + \binom{1011}{1010} 15^1 + \binom{1011}{1011} 1^0$.$(15 + 1)^{1011} = 15 	imes K + 1$,where $K$ is an integer.Therefore,$\frac{4^{2022}}{15} = \frac{15K + 1}{15} = K + \frac{1}{15}$.The fractional part is $\left\{ \frac{4^{2022}}{15} ight\} = \frac{1}{15}$.

The fractional part of the number $\frac{4^{2022}}{15}$ is equal to

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