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 ,
Checking back to the fraction, all of these do indeed yield integers.
Now, returning to and
and
Since , must be . This yields four possible pairs , , ,
See Also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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All AMC 12 Problems and Solutions |