Question

Rodrigo 有一张很大的方格纸。首先，他画一条线段连接点 $(0, 4)$ 与点 $(2, 0)$ ，并给线段内部穿过的 $4$ 个方格涂色，如下图所示。接着，Rodrigo 画一条线段连接点 $(2000, 3000)$ 与点 $(5000, 8000)$ 。他同样给线段内部穿过的方格涂色。这次他会涂多少个方格？

Rodrigo has a very large piece of graph paper. First he draws a line segment connecting point $(0, 4)$ to point $(2, 0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next, Rodrigo draws a line segment connecting point $(2000, 3000)$ to point $(5000, 8000).$ Again he colors the cells whose interiors intersect the segment. How many cells will he color this time?

$6000$

$6500$

$7000$

$7500$

$8000$

答案：C

知识点：最大公约数格点

难度评级：1810

视频讲解：

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文字解答：

若线段水平差为 $a$ 、竖直差为 $b$ ，它穿过 $a$ 条竖直网格线和 $b$ 条水平网格线，但在格点处的穿越会被重复计算。因此内部被线段穿过的方格数为 $a+b-\gcd(a,b).$

这里坐标差为 $3000$ 和 $5000$ ，且 $\gcd(3000,5000)=1000$ 。所以方格数为 $3000+5000-1000=7000.$

正确答案是 C。

For a segment whose endpoint differences are $a$ horizontally and $b$ vertically, the segment crosses $a$ vertical grid lines and $b$ horizontal grid lines, but crossings at lattice points are counted twice. Therefore the number of cells whose interiors are intersected is $a+b-\gcd(a,b).$

Here the endpoint differences are $3000$ and $5000$ , and $\gcd(3000,5000)=1000$ . Thus the number of cells colored is $3000+5000-1000=7000.$

Thus, C is the correct answer.

2024 AMC 8 第 23 题

答案：C

难度评级：1810

视频讲解：

文字解答：

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