Remainder when 64 32 32 is divided by 9 is equal to _____ . [2024]

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Solution

Q.
Remainder when $64^{32^{32}}$ is divided by 9 is equal to _____ . [2024]

Ans.
(1)

We have, $({{(64)}^{32})}^{32} = ({{(8^{2})}^{32})}^{32} = 8^{2048}$

$= {(9 - 1)}^{2048}$

$= 9^{2048} - {}^{2048}{C_{1}} 9^{2047} + \dots + {}^{2048}{C_{2048} = 9 k + 1}$

So, remainder = 1

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