2023 AMC 10B Problem 8

Attempt Problem 8 of the 2023 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10B solutions, or check the answer key.

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What is the units digit of $2022^{2023} + 2023^{2022}?$

$7$

$1$

$9$

$5$

$3$

Answer: A

Concepts:units digit modular arithmetic pattern recognition

Difficulty rating: 1250

Solution:

Only the units digits matter. Powers of $2$ cycle $2, 4, 8, 6$ with period $4,$ and $2023 \equiv 3 \pmod 4,$ so $2022^{2023}$ ends in $8.$ Powers of $3$ cycle $3, 9, 7, 1,$ and $2022 \equiv 2 \pmod 4,$ so $2023^{2022}$ ends in $9.$ Add them: $8 + 9 = 17,$ so the units digit is $7.$ Therefore, the answer is A.

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