AMC8 2014

AMC8 2014 · Q23

AMC8 2014 · Q23. It mainly tests Logic puzzles, Primes & prime factorization.

Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of your two uniform numbers is today's date. What number does Caitlin wear?

欧几里得中学女子垒球队的三名队员进行了以下对话。 Ashley：我刚意识到我们的队服号码都是两位数的质数。 Bethany：而且你们两个号码的和是我本月初生日的日期。 Caitlin：真有趣。你们两个号码的和是我本月晚些时候生日的日期。 Ashley：而你们两个号码的和是今天的日期。 Caitlin 穿的号码是多少？

(A) 11 11

(B) 13 13

(D) 19 19

(E) 23 23

Answer

Correct choice: (A)

正确答案：（A）

Solution

The maximum amount of days any given month can have is $31$, and the smallest two-digit primes are $11, 13,$ and $17$. There are a few different sums that can be deduced from the following numbers, which are $24, 30,$ and $28$, all of which represent the three days. Therefore, since Bethany says that the other two people's uniform numbers are earlier, so that means Caitlin and Ashley's numbers must add up to $24$. Similarly, Caitlin says that the other two people's uniform numbers are later, so the sum must add up to $30$. This leaves $28$ as today's date. From this, Caitlin was referring to the uniform wearers $13$ and $17$, telling us that her number is $11$, giving our solution as $\boxed{(A) 11}$.

一个月最多有 $31$ 天，最小的两位数质数是 $11, 13$ 和 $17$。从这些数字可以推导出几个不同的和，即 $24, 30$ 和 $28$，它们分别表示三个日期。因此，Bethany 说其他两人的号码的和是较早的日期，这意味着 Caitlin 和 Ashley 的号码之和必须是 $24$。同样，Caitlin 说其他两人的号码之和是较晚的日期，所以和必须是 $30$。这留下 $28$ 作为今天的日期。由此可得，Caitlin 指的是号码为 $13$ 和 $17$ 的队员，说明她的号码是 $11$，因此答案是 $\boxed{(A) 11}$。

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