AMC10 2019 A

AMC10 2019 A · Q10

AMC10 2019 A · Q10. It mainly tests Counting in geometry (lattice points), GCD & LCM.

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 last tile, how many tiles does the bug visit?

一个宽 10 英尺、长 17 英尺的矩形地板铺了 170 张一英尺见方的瓷砖。一只虫子从一个角直线走到对角。包括首尾瓷砖在内，虫子经过多少张瓷砖？

(A) 17 17

(B) 25 25

(D) 27 27

(E) 28 28

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Because 10 and 17 are relatively prime, the diagonal does not cross the boundaries between tiles at any corner point of the tiles. In order for the bug to move from one vertex of the rectangle to the opposite vertex, the bug must cross 9 edges in one direction and 16 edges in the other direction, a total of 25 edges. Each time the bug crosses an edge, it enters a new tile. Counting the tile it started on as well, the bug visits a total of $1 + 25 = 26$ tiles.

答案（C）：因为 10 和 17 互质，对角线不会在任何瓷砖的角点处穿过瓷砖之间的边界。为了让虫子从矩形的一个顶点移动到对面的顶点，它必须在一个方向上穿过 9 条边、在另一个方向上穿过 16 条边，总共穿过 25 条边。每当虫子穿过一条边，它就进入一块新的瓷砖。把起始所在的那块瓷砖也算上，虫子一共访问了 $1 + 25 = 26$ 块瓷砖。

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