2023-2023-mod-13-

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

2023^{2023} = \dots (\mod 13)

Answered by Rasheed.Sindhi last updated on 05/Dec/23

2023^{2023} \equiv \dots (\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