AMC10 2021 B

AMC10 2021 B · Q5

AMC10 2021 B · Q5. It mainly tests Casework, Divisibility & factors.

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?

Jonie 的四个堂兄弟的年龄是不同的个位正整数。其中两个堂兄弟的年龄相乘得 $24$，另外两个相乘得 $30$。Jonie 四个堂兄弟年龄的总和是多少？

(A) 21 21

(B) 22 22

(D) 24 24

(E) 25 25

Answer

Correct choice: (B)

正确答案：（B）

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 \[3 + 8 + 5 + 6 = \boxed{\textbf{(B)} ~22}.\]

首先看相乘为 $24$ 的两个堂兄弟年龄。由于年龄必须是个位数，可能的年龄是 $3$ 和 $8$ 或 $4$ 和 $6$。接下来看相乘为 $30$ 的两个堂兄弟年龄。由于年龄必须是个位数，唯一可能的年龄是 $5$ 和 $6$。记住所有年龄必须互不相同，唯一可行的方案是年龄为 $3,8$ 和 $5,6$。要求年龄总和，即 \[3 + 8 + 5 + 6 = \boxed{\textbf{(B)} ~22}\]。

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