AMC 10 · 2025 · #17
Grade 6 number-theoryLet 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?
Pick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Dividing 273436 by N leaves a remainder of 16; Dividing 272760 by N leaves a remainder of 15; N is the unique positive integer that fits both facts
Unknowns: The tens digit of N
Understand
Restated: 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.
Givens: Dividing 273436 by N leaves a remainder of 16; Dividing 272760 by N leaves a remainder of 15; N is the unique positive integer that fits both facts
Plan
Primary tool: #13 Convert to Algebra
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
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.
Execute — Answer: E
4.NBT.B.6 Step 1 Turn remainders into exact division
- If dividing a number by N leaves remainder r, then that number minus r is a multiple of N.
- Subtract each remainder to get numbers N divides with nothing left over.
💡 A remainder is the leftover after fitting whole copies of N, so taking it away leaves an exact stack of N's.
6.NS.B.4 Step 2 N divides their difference
- N divides both 273420 and 272745, so it also divides their difference.
- Subtracting turns the two huge numbers into one small number that N must divide.
💡 If two numbers are both stacks of N, the gap between them is also a whole number of N's.
4.NBT.B.6 Step 3 Find the greatest common divisor
- N must divide both 675 and 272745, so N divides their greatest common divisor.
- Use repeated division (the Euclidean algorithm): 272745 = 675 x 404 + 45, then 675 = 45 x 15 with no remainder, so the gcd is 45.
- Every valid N divides 45.
💡 Chasing the remainders downward with division always lands on the largest number that splits both evenly.
4.OA.B.4 Step 4 Use the remainder rule to pick N
- The divisors of 45 are 1, 3, 5, 9, 15, 45.
- A remainder of 16 is only possible when the divisor is bigger than 16, so N > 16.
- The only divisor of 45 above 16 is 45 itself, so N = 45.
💡 A remainder can never reach the divisor, so a remainder of 16 forces the divisor past 16 and knocks out every small factor.
2.NBT.A.1 Step 5 Read the tens digit
- N = 45.
- Its tens digit is 4, so the answer is (E).
💡 In a two-digit number the left digit counts the tens, so 45 has 4 tens.
4.NBT.B.6 If dividing a number by N leaves remainder r, then that number minus r is a mult 6.NS.B.4 N divides both 273420 and 272745, so it also divides their difference. Subtracti 4.NBT.B.6 N must divide both 675 and 272745, so N divides their greatest common divisor. U 4.OA.B.4 The divisors of 45 are 1, 3, 5, 9, 15, 45. A remainder of 16 is only possible wh 2.NBT.A.1 N = 45. Its tens digit is 4, so the answer is (E). Review
Reasonableness: 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.
Alternative: Skip the difference and factor each number: 273420 = 45 x 6076 and 272745 = 45 x 6061, where 6076 and 6061 differ by 15 and share no factor with each other's structure that exceeds 45. Their common divisors are exactly the divisors of gcd(273420, 272745) = 45, and requiring the divisor to exceed 16 again pins N = 45.
CCSS standards used (min grade 6)
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Rewriting each remainder fact as an exact multiple and running the Euclidean divisions)6.NS.B.4Find greatest common factor and least common multiple of two numbers (Recognizing N as a common divisor and reducing to the gcd of the two multiples)4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Listing the divisors of 45 to test against the size rule)2.NBT.A.1Understand that the three digits of a three-digit number represent hundreds, tens, and ones (Reading the tens digit of 45)
⭐ 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.
⭐ 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.
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