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
Basic knowledge about prime numbers , simple logical reasoning and number theory.
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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