2007 AMC 12B Problems/Problem 24
Problem 24
How many pairs of positive integers are there such that and is an integer?
Solution
Combining the fraction, must be an integer.
Since the denominator contains a factor of ,
Rewriting as for some positive integer , we can rewrite the fraction
Since the denominator now contains a factor of ,
Rewriting as for some positive integer , we can rewrite the fraction again as
Since the denominator contains ,
Now, returning to and
and
Since , must be . This yields four possible pairs , , ,