find-the-last-four-digits-of-2022-2023-2023-2022-

Question Number 203192 by MrGHK last updated on 12/Jan/24

find the last four digits of 2022^{2023} + 2023^{2022}

Answered by AST last updated on 12/Jan/24

x = 2022^{2023} + 2023^{2022} \overset{16}{\equiv} 6^{2023} + 7^{2022} \overset{16}{\equiv} 2^{2023} 3^{2023} + 7^{6}

\overset{16}{\equiv} 7^{6} \overset{16}{\equiv} 1

x = 2022^{2023} + 2023^{2022} \overset{625}{\equiv} {(147)}^{23} + 148^{22} \overset{625}{\equiv}

\overset{625}{\equiv} {(147)}^{26} {(147)}^{- 2} + {(148)}^{24} {(148)}^{- 2}

\equiv 4 {(147)}^{- 2} + 356 {(148)}^{- 2}

\frac{1}{148^{2}} \overset{625}{\equiv} z \Rightarrow 148^{2} z \equiv 29 z \overset{625}{\equiv} 1 \Rightarrow z \overset{625}{\equiv} 194

\Rightarrow x \equiv 4 (372) + 356 (194) \equiv 552 (m o d 625)

x = 625 q + 552 \equiv 1 (m o d 16) \Rightarrow q \equiv 9 (m o d 16)

\Rightarrow x = 625 (16 k + 9) + 552 \Rightarrow x = 10000 k + 6177

Commented by MrGHK last updated on 13/Jan/24

t h a n k s

Commented by MathematicalUser2357 last updated on 15/Jan/24

q + 552

r e a d t h e b l u e o n e