2001 Pan African MO Problems/Problem 1
Find all positive integers such that: is a positive integer.
Perform polynomial long division to get . Note that if , then can not be an integer. Thus, all of the solutions satisfy the inequality .
If , then . However, there are no positive integers in this case. If , then and . Rearranging the second inequality results in . Factoring results in , so .
Now there are only four possible positive integers, so we can use trial and error to determine if is a positive integer. After doing trial and error, the only positive integers that make an integer are or .
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