2021 AMC 10B Problems/Problem 5

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?

$\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~25$

Solution

First look at the two cousins' ages that multiply to $24$. Since the ages must be single-digit, the ages must either be $3 \text{ and } 8$ or $4 \text{ and } 6.$

Next, look at the two cousins' ages that multiply to $30$. Since the ages must be single-digit, the only ages that work are $5 \text{ and } 6.$ Remembering that all the ages must all be distinct, the only solution that works is when the ages are $3, 8$ and $5, 6$.

We are required to find the sum of the ages, which is \[3 + 8 + 5 + 6 = \boxed{(B) \text{ } 22}.\]

-PureSwag

Video Solution by OmegaLearn (Using Factors)

https://youtu.be/oe7lnbDO8bo

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/gLahuINjRzU?t=857

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=358

~Interstigation

See Also

2021 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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