13. The lines $y=x$ and $y=m x-4$ intersect at the point $P$.
What is the sum of the positive integer values of $m$ for which the coordinates of $P$ are also positive integers?
A 3
B 5
C 7
D 8
E 10
SOlUTION
E

At the point $P$ we have both $y=x$ and $y=m x-4$. Therefore at $P$ we have $x=m x-4$ and hence $4=m x-x$. This last equation may be written as $4=(m-1) x$. If $m$ and $x$ are both positive integers which satisfy this last equation, then $m-1$ and $x$ are both positive integers which are factors of 4 .
Hence the possible values of $m-1$ are 1,2 and 4 with corresponding values of 2,3 and 5 for $m$ and 4,2 and 1 for $x$ and $y$.
Hence the sum of the positive integer values of $m$ for which the coordinates of $P$ are positive integers is $2+3+5=10$.