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AMC-8 USA Math Olympiad

Number theory AMC 8 2014 Problem Number 23

Try this beautiful problem from AMC 8. It involves number theory and logical reasoning. We provide sequential hints so that you can try the problem.

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.
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 $31$, 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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