1994 AHSME Problems/Problem 28
Contents
Problem
In the -plane, how many lines whose -intercept is a positive prime number and whose -intercept is a positive integer pass through the point ?
Solution 1
The line with -intercept and -intercept is given by the equation . We are told is on the line so
Since and are integers, this equation holds only if is a factor of . The factors are which means must be one of . The only members of this list which are prime are and , so the number of solutions is .
Solution 2
Let , , and . As stated in the problem, the -intercept is a positive prime number, and the -intercept is a positive integer.
Through similar triangles, , ,
The only cases where is prime are:
So the number of solutions are .
See Also
1994 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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