Competition · AMC preparation · step 4 of 4

AMC 10 · 2021B · #5

Grade 5 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
factorssystematic-enumerationdigit-constraintsmultiples systematic-enumerationcasework ↑ Prerequisites: factors
📏 Short solution 💡 2 insights
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Problem
Jonie has four cousins whose ages are four different single-digit positive integers (so each age is in {1, 2, …, 9}). One pair of ages multiplies to 24, the other pair multiplies to 30. Find the sum of the four ages.

Pick an answer.

(A)
21
(B)
22
(C)
23
(D)
24
(E)
25

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

Tool #2 (Systematic List) — list every single-digit factor pair of 24 in increasing order, then every single-digit factor pair of 30. The set of pairs is tiny, so listing finishes quickly. Tool #3 (Eliminate) handles the "distinct" constraint: combine each 24-pair with each 30-pair, drop any combination that repeats an age, and the survivor is forced.

1STEP 1

List factor pairs of 24

Among single-digit factor pairs of 24, only {3, 8} or {4, 6} stay one-digit.

24 = 3 × 8 = 4 × 6
2STEP 2

List factor pairs of 30

Among single-digit factor pairs of 30, only {5, 6} stays one-digit.

30 = 5 × 6
3STEP 3

Drop the repeated age

Pairing {4, 6} with {5, 6} repeats the 6, so distinctness forces {3, 5, 6, 8}.

{4, 6} ∪ {5, 6} = {4, 5, 6} (repeat) → drop
4STEP 4

Add the four ages

Add the four ages: 3 + 5 + 6 + 8 = 22, which is choice (B).

3 + 5 + 6 + 8 = 22 → (B)
Answer
22
Double-check the two products: 3 × 8 = 24 ✓ and 5 × 6 = 30 ✓. All four ages {3, 5, 6, 8} are distinct single-digit positive integers ✓. Sum 22 lies right in the middle of the answer choices 21–25, which is plausible for four ages around 5–6 each.
💡Key takeaway

This AMC 10 problem only needs Grade 5 "list and compare factor pairs" — 24 = 3 × 8 or 4 × 6, 30 = 5 × 6, and the "all distinct" rule forces {3, 5, 6, 8} with sum 22!

  • List factor pairs of 24
  • List factor pairs of 30
  • Drop the repeated age
  • Add the four ages

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