2007 AMC 10B Problems/Problem 25
How many pairs of positive integers (a,b) are there such that a and b have no common factors greater than 1 and:
is an integer?
Contents
Solution
Solution 1
Getting common denominators, we have to find coprime such that . Clearly, . Since and are coprime, . Similarly, . However, cannot be as only has solutions when . Therefore, and . Checking them all (Or noting that is the smallest answer choice), we see that they work and the answer is .
Solution 2
Let . We can then write the given expression as where is an integer. We can rewrite this as a quadratic, . By the Quadratic Formula, . We know that must be rational, so must be a perfect square. Let . Then, . The factors pairs of are and , and , and , and and . Only and and and give integer solutions, and and and , respectively. Plugging these back into the original equation, we get possibilities for , namely and .
See Also
2007 AMC 10B (Problems • Answer Key • Resources) | ||
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