2014 AMC 8 Problem 23 Number Theory

Understand the problem

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. Brittany : 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? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$
Source of the problem
2014 amc 8  problem 23
Topic
Number Theory
Difficulty Level
Easy
Suggested Book
Mathematical Circles

Start with hints

Do you really need a hint? Try it first!

The maximum amount of days any given month can have is 31 . 2 digit primes are like to be 11 ,13 …  
There are many two digit primes and three primes will be choosen in such a way that sum of any two of them must not exceed 31 . Proceed to find such three .
Certainly these are 11 , 13 and  17 respectively . All three distinct sums are 24  ,  28  and  30 respectively  . Proceed to select the correct one for  Caitlin .
Since Brittany says “And the sum of your two uniform numbers is the date of my birthday earlier this month .”  So that means Caitlin and Ashley’s numbers must add up to 24.    \( \\ \\ \) Similarly, Caitlin says “The sum of your two uniform numbers is the date of my birthday later this month.”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 .

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