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Date: 01/07/98 at 03:21:18
Subject: Algebra

A basketball court is made up of square parquet tiles, all the same
size, laid side by side to form a rectangle 105 tiles wide and 135
tiles long. If a straight line is drawn diagonally from one corner of
the floor to the opposite corner, how many tiles will the diagonal
intersect?

Thanks.
```

```
Date: 01/07/98 at 06:21:50
From: Doctor Mitteldorf
Subject: Re: Algebra

Dear Arash,

Let's divide the court into 15 lengthwise and 15 widthwise, so that
each section is 7 by 9 tiles. If you draw a diagonal across the whole
thing, it cuts exactly through the corner of each 7x9 rectangle, so it
goes through 15 corners like that. I don't know if you want to call
this going through two tiles or going through none, when the line
passes exactly through a corner. For now, I will call it not going
through any tiles at all.

Now, within each 7x9 rectangle, the line must pass 8 sides to go the
length and 6 sides to go the width. Each time it goes through a side,
it enters another tile. Counting the tile it started in as 1, it must
pass through 1+8+6 = 15 tiles to get across the 7x9 rectangle. Since
there are 15 rectangles, and the line goes through 15 tiles to pass
through each rectangle, the answer is 15*15 = 225 tiles that the line
passes through. (There are another 15 pairs of tiles that it "touches"
as it passes by the corner only.)

-Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 01/15/98 at 23:02:55
From: Doctor Rob
Subject: Re: Algebra

Since 105 and 135 have the common divisor 15, the diagonal will
cross 15 times as many tiles as the diagonal of a 105/15 = 7 by
135/15 = 9 tile rectangle.

Now since 7 and 9 have no common factor, the answer for the small
rectangle is 7 + 9 - 1 = 15. The final answer is then 15*15 = 225.

I leave it to you to prove that if x and y have no common factor, then
the diagonal of an x-by-y rectangle intersects x + y - 1 tiles.

HINT:  Start at one corner with one square. What happens every time
the diagonal crosses either a horizontal or vertical line? Figure
out why we need x and y to have no common factor.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
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