AMC 10 · 2025 · #17

Grade 6 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
gcdmodular-arithmeticdivisibility-rulesfactors bound-inequality-then-enumerate ↑ Prerequisites: gcddivisibility-rules
📏 Medium solution 💡 3 insights
Problem
One positive whole number N works for both of these: dividing 273436 by N leaves remainder 16, and dividing 272760 by N leaves remainder 15. Find the tens digit of that N.

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
3
(E)
4

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

The words 'leaves a remainder' are hard to use directly, so first rewrite them as clean divisibility facts: subtract each remainder to get a number N divides exactly. Once N divides two numbers, it divides their difference, which shrinks the giant numbers down to a small one. Finding the largest possible N becomes a greatest-common-divisor subproblem, and then the rule N ＞ 16 eliminates all but one divisor.

1STEP 1

Turn remainders into exact division

Remainder r means the number minus r is a multiple of N, so subtract: 273436 - 16 = 273420 and 272760 - 15 = 272745.

N ∣ (273436 - 16) = 273420, N ∣ (272760 - 15) = 272745
2STEP 2

N divides their difference

N divides both of those, so it divides their difference: 273420 - 272745 = 675.

N ∣ (273420 - 272745) = 675
3STEP 3

Find the greatest common divisor

Euclid on 272745 and 675: 272745 = 675 x 404 + 45, then 675 = 45 x 15, so the gcd is 45 and N divides it.

gcd(272745, 675) = gcd(675, 45) = 45
4STEP 4

Use the remainder rule to pick N

Divisors of 45 are 1, 3, 5, 9, 15, 45; remainder 16 forces N greater than 16, so N = 45.

divisors of 45: 1, 3, 5, 9, 15, 45; N ＞ 16 → N = 45
5STEP 5

Read the tens digit

N = 45. Its tens digit is 4, so the answer is (E).

45 = 4 tens + 5 ones → tens digit = 4
Answer
4
Check N = 45 directly. 273420 / 45 = 6076 exactly, so 273436 = 45 x 6076 + 16, remainder 16. And 272745 / 45 = 6061 exactly, so 272760 = 45 x 6061 + 15, remainder 15. Both original facts hold, and 45 ＞ 16 so a remainder of 16 is legal. N = 45 is the unique answer, and its tens digit is 4.
💡Key takeaway

Subtract each remainder to get exact multiples, take their difference and its gcd to trap N, then use 'remainder smaller than divisor' to pick the one that fits.

  • Turn remainders into exact division
  • N divides their difference
  • Find the greatest common divisor
  • Use the remainder rule to pick N
  • Read the tens digit

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