1989 AMC 12 Problem 16

Attempt Problem 16 of the 1989 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1989 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

16.

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.)

2

4

6

16

46

Compute the horizontal and vertical coordinate differences

A segment with differences

\Delta x,\Delta y

contains

\gcd(|\Delta x|,|\Delta y|)+1

lattice points

The coordinate differences are

45

and

264,

whose greatest common divisor is

3.

The number of lattice points, including both endpoints, is therefore

\gcd(45,264)+1=4.

Thus the correct answer is B.

Problem 16 in other years