2019 AMC 10A Problem 10

Question

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

Answer: C

Concepts:greatest common divisor counting intersections

Difficulty rating: 1240

Small hint:

Count every grid line the diagonal crosses

Big hint:

Because

10

and

17

are relatively prime, it never passes through an interior grid corner

Solution:

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.