2021 AMC 10B Problem 5

P1-10~2 minPrint

Divisibility & Factors Organized listing / direct enumeration

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?

Show solution without answering

Key insight

Using single digits, 30 can only be 5 × 6. Since 6 is now taken, 24 has to be 3 × 8, not 4 × 6.

Solution

Each product is made from two different ages, each from $1$ to $9$ . List the ways to make each product.

Start with 30, since it has fewer choices:

$1 \times 30$	$2 \times 15$	$3 \times 10$	$5 \times 6$

Only $5 \times 6$ uses two single digits. So two cousins are $5$ and $6$ .

Now 24:

$1 \times 24$	$2 \times 12$	$3 \times 8$	$4 \times 6$

The single-digit options are $3 \times 8$ and $4 \times 6$ . But $6$ is already one cousin's age, and all four ages are different. So the other two cousins are $3$ and $8$ .

Add all four ages: $5 + 6 + 3 + 8 = 22$ .

The answer is $\boxed{\textbf{(B)}\ 22}$ .

Why this works

Start with the part that has the fewest possibilities. The single-digit rule leaves $30$ only one choice, and then "all ages different" leaves $24$ only one choice too. A short list plus one rule is quicker than any algebra.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Write each product as a pair of distinct single-digit factors.

$30 = 1 \cdot 30 = 2 \cdot 15 = 3 \cdot 10 = 5 \cdot 6$ ; only $5 \cdot 6$ uses single digits. So two cousins are $5$ and $6$ .

$24 = 3 \cdot 8 = 4 \cdot 6$ (the pairs $1 \cdot 24$ and $2 \cdot 12$ are out). Since $6$ is already used and ages are distinct, the other two cousins are $3$ and $8$ .

Sum: $3 + 8 + 5 + 6 = 22$ .

The answer is $\boxed{\textbf{(B)}\ 22}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics (§1.4, §1.5)
Prealgebra, Chapter 15: Problem-Solving Strategies (§15.2)
Introduction to Number Theory, Chapter 5: Divisor Problems (§5.2, §5.3)
Introduction to Number Theory, Chapter 13: Divisibility Rules (§13.2)