The remainder when is divided by 7 is equal to [2025]

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Solution

Q.
The remainder when $({{(64)}^{(64)})}^{(64)}$ is divided by 7 is equal to [2025]

1 3

2 1

3 6

4 4

Ans.
(2)

Let $A = ({{(64)}^{(64)})}^{(64)} = {(64)}^{64^{2}}$

$A = {(1 + 63)}^{64^{2}}$ , let $64^{2} = n$

Expanding by binomial

$A = {(1 + 63)}^{n} = 1 + {}^{n}C_{1} (63) + {}^{n}C_{2} {(63)}^{2} + . . . = 1 + 63 λ = 1 + 7 (9 λ)$

$∴$ When $({{(64)}^{(64)})}^{(64)}$ is divisible by 7 then remainder is 1.

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