AMC10 2024 B

AMC10 2024 B · Q7

AMC10 2024 B · Q7. It mainly tests Remainders & modular arithmetic, Powers & residues.

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

$7^{2024}+7^{2025}+7^{2026}$ 除以 $19$ 的余数是多少？

(A) 0 0

(B) 1 1

(D) 11 11

(E) 18 18

Answer

Correct choice: (A)

正确答案：（A）

Solution

We can factor the expression as \[7^{2024} (1 + 7 + 7^2) = 7^{2024} (57).\] Note that $57=19\cdot3$, this expression is actually divisible by 19. The answer is $\boxed{\textbf{(A) } 0}$.

我们可以因式分解该表达式： \[7^{2024} (1 + 7 + 7^2) = 7^{2024} (57)。\] 注意 $57=19\cdot3$，该表达式实际上能被19整除。答案为 $\boxed{\textbf{(A) } 0}$。

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