1994 AHSME Problems/Problem 28

Problem

In the $xy$ -plane, how many lines whose $x$ -intercept is a positive prime number and whose $y$ -intercept is a positive integer pass through the point $(4,3)$ ?

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

Let the line be $y=mx+c$ , with $c$ being a positive integer. Then we have $3=4m+c$ so $m=(3-c)/4$ . Now the $x$ -intercept is $-c/m = -4c/(3-c) = 4c/(c-3) = (4c-12+12)/(c-3) = 4+12/(c-3).$

We need the $x$ -intercept to be positive, so $4c/(c-3)>0$ and $c>0$ together imply $c-3>0 \implies c>3$ , so if the $x$ -intercept is to be an integer, $(c-3)$ must be a positive factor of $12$ . Hence we can get $4+1$ , $4+2$ , $4+3$ , $4+4$ , $4+6$ , and $4+12$ , and the only ones of these that are prime are $4+1=5$ and $4+3=7$ .

The first case gives $c-3=12 \implies c=15$ and the second case gives $c-3=4 \implies c=7$ , and both of these satisfy all the conditions of the problem, so the number of solutions is $\boxed{\textbf{(C) } 2}$ .

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1994 AHSME Problems/Problem 28

Problem

Solution