2023^(2023) = ... (mod 13)


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Question Number 201352 by cortano12 last updated on 05/Dec/23

$2023^{2023} = . . . (\mod 13)$
	Answered by Rasheed.Sindhi last updated on 05/Dec/23

	$2023^{2023} \equiv . . . (\mod 13)$ $2023^{2023}$ $\equiv 8^{2023} \equiv x (\mod 13) [∵ 2023 \equiv 8 (\mod 13)]$ $∵ 8^{4} \equiv 1 (m o d 13$ $∴ 8^{2023} = {(8^{4})}^{505} (8^{3}) \equiv 8^{3} \equiv 5 (\mod 13)$
	Answered by mr W last updated on 05/Dec/23

	$2023^{2023} m o d 13$ $= {(155 \times 13 + 8)}^{2023} m o d 13$ $\equiv 8^{2023} m o d 13$ $= 8 \times {(64)}^{1011} m o d 13$ $= 8 \times {(5 \times 13 - 1)}^{1011} m o d 13$ $\equiv - 8 m o d 13$ $\equiv 5 m o d 13$
	Answered by BaliramKumar last updated on 05/Dec/23

	$2023^{2023} = x (\mod 13) [ϕ (13) = 12]$ ${(13 \times 155 + 8)}^{(12 \times 168 + 7)} = x (\mod 13)$ ${(8)}^{(7)} = x (\mod 13)$ ${(8^{2})}^{3} \times 8^{1} = x (\mod 13)$ ${(64)}^{3} \times 8^{1} = x (\mod 13)$ ${(- 1)}^{3} \times 8^{1} = x (\mod 13)$ $- 8 = x (\mod 13)$ $1 \times 13 - 8 = x (\mod 13)$ $5 = 5 (\mod 13)$ $x = 5$
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