Problem:
Elmo makes sandwiches for a fundraiser. For each sandwich he uses globs of peanut butter at cents per glob and blobs of jam at cents per blob. The cost of the peanut butter and jam to make all the sandwiches is . Assume that , , and are positive integers with . What is the cost of the jam Elmo uses to make the sandwiches?
Answer Choices:
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Solution:
The total cost of the peanut butter and jam is cents, so and are factors of . Because , the possible values of are , and . If , then , which is impossible since and are positive integers. If , then , which also has no solutions in positive integers. Hence and , which has the unique positive integer solution and . So the cost of the jam is .
The problems on this page are the property of the MAA's American Mathematics Competitions