2025 AMC 10A Problem 17

Below is the professionally curated solution for Problem 17 of the 2025 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 10A solutions, or check the answer key.

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Concepts:greatest common divisor divisibility

Difficulty rating: 1730

17.

Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16,$ and dividing $272760$ by $N$ leaves a remainder of $15.$ What is the tens digit of $N?$

$0$

$1$

$2$

$3$

$4$

Solution:

Subtract off the remainders. Both $273420 = 273436 - 16$ and $272745 = 272760 - 15$ are multiples of $N,$ so their difference $675$ is too. Now $272745 = 404 \cdot 675 + 45,$ which makes $45$ a multiple of $N$ as well. The remainder $16$ means $N \gt 16,$ and the only divisor of $45$ bigger than $16$ is $45$ itself. So $N = 45,$ and its tens digit is $4.$ Thus, E is the correct answer.

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