Question

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?

$11$

$13$

$17$

$19$

$23$

Solution:

Any pair-sum must be a date in the month, so it is at most $31$ . The possible sums of two two-digit primes that are at most $31$ are $11+13=24$ , $11+17=28$ , $11+19=30$ , and $13+17=30$ .

The three dates are earlier, today, and later, so they must be distinct and increasing. The only possible set of three distinct dates is $24,28,30$ , coming from the numbers $11$ , $13$ , and $17$ .

Bethany's date is Ashley plus Caitlin, Ashley's date is Bethany plus Caitlin, and Caitlin's later date is Ashley plus Bethany. This means Caitlin's number is the smallest of the three, namely $11$ .

Thus, A is the correct answer.

2014 AMC 8 Problem 23

Solution:

Problem 23 in Other Years