2012 AMC 10A Problems/Problem 17
Let and be relatively prime integers with and = . What is ?
Since and are both integers, so must and . For this fraction to simplify to , the denominator, or , must be a multiple of 3. Looking at the answer choices, it is only possible when .
Using difference of cubes in the numerator and cancelling out one in the numerator and denominator gives .
Set , and . Then . Cross multiplying gives , and simplifying gives . Since and are relatively prime, we let and , giving and . Since , the only solution is , which can be seen upon squaring and summing the various factor pairs of .
An alternate method of solving the system of equations involves solving the second equation for , plugging it into the first equation, and solving the resulting quartic equation with a substitution of . The four solutions correspond to
Thus, the desired quantity .
Note that if you double and double , you will get different (but not relatively prime) values for and that satisfy the original equation.
Or we can solve for directly instead of solving for and . ,
The first step is the same as above which gives .
Then we can subtract and then add to get , which gives . . Cross multiply . Since , take the square root. . Since and are integers and relatively prime, is an integer. is a multiple of , so is a multiple of . Therefore and is a solution. So
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