2024 AMC 10B Problem 7

Problem 7 of 25EasierAlgebra Number Theory

What is the remainder when

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

is divided by

19?

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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.

Concepts: modular arithmetic · factoring · exponent

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.