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 $\boxed{\textbf{(C) } 2}$ .

1994 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 27	Followed by Problem 29
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1994 AHSME Problems/Problem 28

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