2024/25 AHSMC Part 1 Problem 7

Problem 7 of 16

A line passes through the points $(3,1)$ and $(k, k+8)$ , where $k$ is a positive integer. If the slope of the line is an integer, how many possible values can $k$ take?

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Solution

The slope is $m = \dfrac{k+7}{k-3} = 1 + \dfrac{10}{k-3}$ . Since $m,k \in \mathbb{Z}$ , the integer $k-3$ divides $10$ , so $k-3 \in \{\pm 1, \pm 2, \pm 5, \pm 10\}$ . As $k$ is positive we need $k-3 \ge -2$ , hence $k-3 \in \{-2,-1,1,2,5,10\}$ , giving $6$ values. The answer is (C).

University of Alberta

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