# 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 -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.

##### Difficulty Level

6/10

##### Suggested Book

Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

# Start with hints

Do you really need a hint? Try it first!

The maximum amount of days any given month can have is , 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 and , 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 . Similarly, Caitlin says that the other two people’s uniform numbers is later, so the sum must add up to . This leaves as today’s date.

From this, Caitlin was referring to the uniform wearers and , telling us that her number is , giving our solution as 11.

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## One reply on “Number theory AMC 8 2014 Problem Number 23”

Caitlins number is 11

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