Question

一个 $10$ 英尺宽、 $17$ 英尺长的矩形地板铺有 $170$ 块一英尺见方的方砖。一只虫子从一个角沿直线走到对角。包括第一块和最后一块方砖在内，它经过多少块方砖？

A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?

$17$

$25$

$26$

$27$

$28$

答案：C

知识点：最大公约数交点计数

难度评级：1240

解答：

每穿过一条内部网格线，虫子就进入一块新方砖。

方砖数要从第一块 $1$ 开始计算。

因为 $10$ 与 $17$ 互质，对角线不会经过内部格点，所以不会同时穿过一条竖线和一条横线。

它穿过 $9$ 条竖直内部线和 $16$ 条水平内部线，因此经过的方砖数为。 $1 + 16 + 9 = 26$

所以正确答案是 C。

Note that every time the bug crosses a vertical or horizontal line, the bug visits one new tile.

This means that the number of tiles the bug visits is $1$ (the first tile) plus the number of lines it crosses.

The bug never walks over a corner since $10$ and $17$ are relatively prime, so we don't have to worry about that.

The bug crosses $16$ horizontal lines and $9$ vertical lines for a total of $1 + 16 + 9 = 26$ tiles.

Thus, C is the correct answer.

2019 AMC 10A 第 10 题

答案：C

难度评级：1240

解答：

其他年份的第 10 题