2025 AMC 10A Problem 17

Problem 17 of 25IntermediateNumber Theory

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?

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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.

More practice

Concepts: greatest common divisor · divisibility

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