1989 AHSME Problems/Problem 16

Problem

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$ ? (Include both endpoints of the segment in your count.)

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46$

Solution

The difference in the $y$ -coordinates is $281 - 17 = 264$ , and the difference in the $x$ -coordinates is $48 - 3 = 45$ . The gcd of 264 and 45 is 3, so the line segment joining $(3,17)$ and $(48,281)$ has slope $\frac{88}{15}.$ The points on the line have coordinates $\left(3+t,17+\frac{88}{15}t\right).$ If $t$ is an integer, the $y$ -coordinate of this point is an integer if and only if $t$ is a multiple of 15. The points where $t$ is a multiple of 15 on the segment $3\leq x\leq 48$ are $3$ , $3+15$ , $3+30$ , and $3+45$ . There are 4 lattice points on this line. Hence the answer $\textbf{(B)}$ .

1989 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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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1989 AHSME Problems/Problem 16

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