AMC 10 · 2021 · #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

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

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

24 = 3 × 8 = 4 × 6

2STEP 2

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

30 = 5 × 6

3STEP 3

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: 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!

More like this