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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

What is the remainder when

7^{2024} + 7^{2025} + 7^{2026}

is divided by

19?

0

1

7

11

18

Factor out

7^{2024}

The remaining factor is

1 + 7 + 49;

check its relationship to

19

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.

Problem 7 in other years