AMC 8 · 2001 · #2

Grade 4 arithmeticnumber-theory
factorssystematic-enumerationmulti-digit-arithmetic systematic-enumerationguess-and-check ↑ Prerequisites: factorsmulti-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem
Two whole numbers have product 24 and sum 11. Find the larger of the two numbers.

Pick an answer.

(A)
3
(B)
4
(C)
6
(D)
8
(E)
12

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a List

Because the product is 24, the two numbers must be a factor pair of 24. There are only four positive whole-number factor pairs, so Tool #2 (Make a List) lets us write them all down in a few seconds. For each pair, Tool #6 (Guess and Check) does the rest: just add the two numbers and see whether the sum is 11. No algebra is needed — the short list does all the work.

1STEP 1

List every positive factor pair of 24, walking the divisors up from 1.

24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6
2STEP 2

Add each pair; only 3 and 8 sums to 11.

pair & sum & equals 11? ; (1, 24) & 25 & no ; (2, 12) & 14 & no ; (3, 8) & 11 & yes ; (4, 6) & 10 & no
3STEP 3

The question asks for the larger of the two, which is 8.

max(3, 8) = 8 → (D)
Answer
8
Verify both conditions on the chosen pair: 3 × 8 = 24 matches the product, and 3 + 8 = 11 matches the sum. Both hold, so 8 is correct. Choice (E) 12 can be ruled out quickly because its partner under product 24 is 2, and 2 + 12 = 14 ≠ 11. Choices (A) 3 and (D) 8 name the two members of the right pair; the question asks for the larger, so (D) wins over (A).
💡Key takeaway

When a problem gives both the product and the sum of two whole numbers, list the factor pairs of the product first — there are usually only a handful — and pick the one whose sum matches. Here the factor pair (3, 8) adds to 11, so the larger number is 8, choice (D).