1966 AHSME Problems/Problem 26

Problem

Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then m can be:

$\text{(A) 4 only} \quad \text{(B) 5 only} \quad \text{(C) 6 only} \quad \text{(D) 7 only} \\ \text{(E) one of the integers 4,5,6,7 and one other positive integer}$

Solution

Substitute the second equation into the first one, we have $13x+11mx-11=700$ .

So $(13+11m)x=711$ . So $13+11m$ is a factor of $711$ . $711=3*3*79$ , so the factors of $711$ are: $1, 3, 9, 79, 237, 711$ .

Clearly, because $m\ge1$ , so $13+11m\ge24$ . So we only need to check whether $m$ is an integer when $11m+13=79, 237, 711$ .

When $11m+13=79$ , $m=6$ .

Checking the other two choices, $m$ dosen't yield to be an integer. So $m=6$ is the only option. Select $\boxed{C}$ .

~hastapasta

1966 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 25	Followed by Problem 27
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1966 AHSME Problems/Problem 26

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