If 25^ 15 is divided by 13, the reminder is _________. - Vidyadip

Question

If $25^{15} $is divided by $13,$ the reminder is _________.

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Answer

If $25^{15}$ is divided by $13,$ the reminder is $12.$
$25^{15}=(26-1)^{15}$ $=\ ^{15}\text{C}_0{26}^{15}-\ ^{15}\text{C}_126^{14}+...+\ ^{15}\text{C}_{14}26-\ ^{15}\text{C}_{15}$
$=(^{15}\text{C}_026^{15}-\ ^{15}\text{C}_126^{14}+...+\ ^{15}\text{C}_{14}26-13)+12$
Clearly$,$ when $25^{15}$ is divided by $13,$ then remainder will be $12.$

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