what is remainder when 61^(61) divided by 1001


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Question Number 104139 by bobhans last updated on 19/Jul/20

$w h a t i s r e m a i n d e r w h e n 61^{61} d i v i d e d$ $b y 1001$
Commented by aurpeyz last updated on 19/Jul/20

$how ?$
	Answered by floor(10²Eta[1]) last updated on 19/Jul/20

	$x \equiv 61^{61} (\mod 1001)$ $61^{2} = 3721 \equiv 718 (\mod 1001)$ $61^{4} \equiv 718^{2} = 515524 \equiv 9 (\mod 1001)$ $61 = 4.15 + 1 \Rightarrow 61^{61} = {(61^{4})}^{15} . 61 \equiv 9^{15} . 61 (\mod 1001) ★$ $9^{4} = 6561 \equiv 555 (\mod 1001)$ $9^{5} \equiv 555.9 = 4995 \equiv - 10 (\mod 1001)$ $9^{15} = {(9^{5})}^{3} \equiv - 1000 \equiv 1 (\mod 1001)$ $★ 9^{15} . 61 \equiv 61 (\mod 1001)$ $\Rightarrow x = 61$
	Answered by OlafThorendsen last updated on 19/Jul/20

	$61^{61} = 61 . {(61^{2})}^{30}$ $61^{61} = 61 . 3721^{30}$ $61^{61} \equiv 61 . 718^{30} [1001]$ $61^{61} \equiv 61 {(718^{2})}^{15} [1001]$ $61^{61} \equiv 61 \times {(9)}^{15} [1001]$ $61$
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