1994 AHSME Problems/Problem 28
In the -plane, how many lines whose -intercept is a positive prime number and whose -intercept is a positive integer pass through the point ?
Let the line be , with being a positive integer. Then we have so . Now the -intercept is
We need the -intercept to be positive, so and together imply , so if the -intercept is to be an integer, must be a positive factor of . Hence we can get , , , , , and , and the only ones of these that are prime are and .
The first case gives and the second case gives , and both of these satisfy all the conditions of the problem, so the number of solutions is .