AMC 10 · 2021 · #5

Grade 5 arithmetic

Archetype Small-Domain Constrained Search

factorssystematic-enumerationdigit-constraintsmultiples systematic-enumerationcasework ↑ Prerequisites: factors

📏 Short solution 💡 2 insights

Problem

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$ , while the other two multiply to $30$ . What is the sum of the ages of Jonie's four cousins?

Pick an answer.

(A)

~21

(B)

~22

(C)

~23

(D)

~24

(E)

~25

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Toolkit + CCSS Solution

Understand

Restated: Jonie has four cousins whose ages are four different single-digit positive integers (so each age is in $\{1, 2, \dots, 9\}$). One pair of ages multiplies to $24$, the other pair multiplies to $30$. Find the sum of the four ages.

Givens: Four ages are distinct positive integers in $\{1, 2, \dots, 9\}$; Two ages multiply to $24$; The other two ages multiply to $30$; Answer choices: (A) $21$, (B) $22$, (C) $23$, (D) $24$, (E) $25$

Unknowns: The sum of the four ages

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #3 Eliminate Possibilities

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.

Execute — Answer: B

#2 Make a Systematic List 4.OA.B.4 Step 1

List the single-digit factor pairs of $24$.
Walk from the smallest factor up: $1 \times 24$ (drop, $24$ is two digits), $2 \times 12$ (drop), $3 \times 8$ (keep), $4 \times 6$ (keep).
So the $24$-pair is $\{3, 8\}$ or $\{4, 6\}$.

$$24 = 3 \times 8 = 4 \times 6$$

💡 List factor pairs in order — Grade 4 "find all factor pairs of a whole number".

#2 Make a Systematic List 4.OA.B.4 Step 2

List the single-digit factor pairs of $30$.
$1 \times 30$ (drop), $2 \times 15$ (drop), $3 \times 10$ (drop), $5 \times 6$ (keep).
Only $\{5, 6\}$ works.

$$30 = 5 \times 6$$

💡 Same factor-pair walk, but here only one pair stays single-digit.

#3 Eliminate Possibilities 5.OA.B.3 Step 3

Combine the $24$-pair with $\{5, 6\}$ and drop the case with a repeat.
Trying $\{4, 6\}$ with $\{5, 6\}$ repeats the $6$ — drop.
Trying $\{3, 8\}$ with $\{5, 6\}$ gives four distinct ages $\{3, 5, 6, 8\}$ — keep.

$$\{4, 6\} \cup \{5, 6\} = \{4, 5, 6\} \text{ (repeat)} \;\Rightarrow\; \text{drop}$$

💡 "Distinct" forces $6$ out of one side — pick the side that survives.

#2 Make a Systematic List 2.NBT.B.5 Step 4

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

$$3 + 5 + 6 + 8 = 22 \;\Rightarrow\; \textbf{(B)}$$

💡 Add four small whole numbers — Grade 2 within-$100$ addition.

[1] #2 4.OA.B.4 List the single-digit factor pairs of $24$. Walk from the smallest factor up: $1

[2] #2 4.OA.B.4 List the single-digit factor pairs of $30$. $1 \times 30$ (drop), $2 \times 15$

[3] #3 5.OA.B.3 Combine the $24$-pair with $\{5, 6\}$ and drop the case with a repeat. Trying $\

[4] #2 2.NBT.B.5 Add the four ages. $3 + 5 + 6 + 8 = 22$, which is choice (B).

Review

Reasonableness: Double-check the two products: $3 \times 8 = 24$ ✓ and $5 \times 6 = 30$ ✓. 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.

Alternative: Tool #6 (Guess and Check) — guess the sum directly. The smallest plausible pair is $\{3, 8\}$ summing $11$, and $\{5, 6\}$ summing $11$, so the total is $11 + 11 = 22$ without listing all cases. Same answer (B).

CCSS standards used (min grade 5)

2.NBT.B.5 Fluently add and subtract within 100 (Computing $3 + 5 + 6 + 8 = 22$ at the end.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Listing every single-digit factor pair of $24$ and of $30$.)
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships (Combining each $24$-pair with the $30$-pair and applying the "all distinct" rule to pick the valid set.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 5)

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