2011 AMC 12A Problems/Problem 7
A majority of the students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was . What was the cost of a pencil in cents?
The total cost of the pencils can be found by .
Since is the product of three sets of values, we can begin with prime factorization, since it gives some insight into the values: . Since neither nor are any of these factors, they can be eliminated immediately, leaving , , and .
Beginning with , we see that the number of pencils purchased by each student must be either or . However, the problem states that the price of each pencil must exceed the number of pencils purchased, so we can eliminate this.
Continuing with , we can conclude that the only case that fulfils the restrictions are that there are students who each purchased such pencils, so the answer is . We can apply the same logic to as we applied to if one wants to make doubly sure.
https://www.youtube.com/watch?v=6tlqpAcmbz4 ~Shreyas S
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