1989 AMC 12 第 16 题

Question

平面上坐标均为整数的点称为格点。端点为

(3,17)

和

(48,281)

的线段上有多少个格点？（计数时包括线段的两个端点。）

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

求出横坐标之差和纵坐标之差

Compute the horizontal and vertical coordinate differences

横、纵坐标之差分别为

\Delta x,\Delta y

的线段上共有

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

个格点

A segment with differences

\Delta x,\Delta y

contains

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

lattice points

横坐标与纵坐标之差分别为

45

和

264

，它们的最大公因数为

3

。因此，包括两个端点在内，格点数为

\gcd(45,264)+1=4

。

所以正确答案是 B。

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.

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