2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 17 Easy

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?

  • A.

    0

  • B.

    1

  • C.

    2

  • D.

    3

  • E.

    4

Answer:E

Since N is unique, we know that N=\gcd(273436-16,272760-15)=\gcd(273420,272745)

Therefore, N \mid (273420 - 272745) = 675 = 5^2 \times 3^3.

Since N \mid \gcd(272745, 675) and noting 25 \nmid 272745, 27 \nmid 272745, we deduce N \mid 5 \times 3^2 = 45. As 45\mid 272745, 675, we have N=45.

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