 # What are we learning ?

Competency in Focus: Number theory This problem from American Mathematics contest (AMC 8, 2014) is based on basic knowledge about prime numbers and simple logical reasoning and number theory .

# First look at the knowledge graph. # Next 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?
##### Source of the problem
American Mathematical Contest 2014, AMC 8 Problem 23
##### Key Competency
Basic knowledge about prime numbers , simple logical reasoning and number theory.
6/10
##### Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

Do you really need a hint? Try it first!
The maximum amount of days any given month can have is $31$, and the smallest two digit primes are 11,13,17. Now see that 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. So,the uniform numbers must be 11,13,17 otherwise there different sums will be more than 31 which gives a contradiction as the maximum amount of days any given moth can have is 31.
Now we have to find which member has what uniform numbers from the given  conversation .
since Brittany 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 is 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 11.

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