2015 AMC 10A Problems/Problem 15
Consider the set of all fractions , where and are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by , the value of the fraction is increased by ?
You can create the equation
Cross multiplying and combining like terms gives .
This can be factored into .
Either has to be positive and has to be negative or vice versa, but if as negative, then would be negative, so that halves our search space.
Similarly, , so and , so .
This leaves the factor pairs: .
But we can't stop here because and must be relatively prime.
gives and . and are not relatively prime so this doesn't work.
gives and . This doesn't work.
gives and . This does work.
We found one valid solution so the answer is .