2024 AMC 10B Problem 7

Attempt Problem 7 of the 2024 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 2024 AMC 10B solutions, or check the answer key.

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What is the remainder when $7^{2024} + 7^{2025} + 7^{2026}$ is divided by $19?$

$0$

$1$

$7$

$11$

$18$

Answer: A

Concepts:modular arithmetic factoring exponent

Difficulty rating: 1250

Solution:

Pull out the common power: $7^{2024} + 7^{2025} + 7^{2026}$ $= 7^{2024}(1 + 7 + 49)$ $= 7^{2024} \cdot 57.$ And $57 = 3 \cdot 19,$ so the product is a multiple of $19.$ The remainder is $0.$ Thus, A is the correct answer.

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