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### Diagonals and Tiles

```
Date: 11/17/2001 at 22:07:53
From: Kamie Oda
Subject: Diagonal and Relatively Prime

Dear Dr. Math,

Jay tiled a 15 feet by 21 feet rectangular ballroom with one-foot-
square tiles. When he finished, he drew both diagonals
connecting opposite corners of the room. What is the total number
of tiles that the diagonals passed through?

I used a procedure from a similar problem from the archives, but
got the wrong answer. Can you help me find my mistake? 15x21,
greatest common divisor is 3, 5+7-1 = 11, 11x3 = 33, 33x 2 diagonals
= 66 tiles. According to my book, the answer is 63. How can the
answer be an odd number when there are 2 diagonals?

Thanks for the help!
Kamie
```

```
Date: 11/18/2001 at 12:29:20
From: Doctor Sarah
Subject: Re: Diagonal and Relatively Prime

Hi Kamie - thanks for writing to Dr. Math.

You need to be careful not to double-count tiles. Let's look at a
diagram:

Tiles with diagonals passing through them are shaded.  Do you
see any tiles with more than one diagonal?  Those tiles should
only be counted once.

Using Relative Primes
http://mathforum.org/dr.math/problems/chin12.7.96.html

- Doctor Sarah, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 11/18/2001 at 15:06:18
From: Marlene Oda
Subject: Re: Diagonal and Relatively Prime

Dear Doctor Sarah,

Thank you for clearly showing me where my extra tiles came
from. I had tried drawing diagonals on graph paper, but my drawing
wasn't accurate and I failed to see the overlap. Thank you for
taking the time to provide the illustration, which clearly showed
3 overcounted tiles! Therefore the answer is 63 not 66. How
enlightening!

Kamie
```
Associated Topics:
High School Basic Algebra
High School Euclidean/Plane Geometry
High School Geometry
High School Puzzles
High School Triangles and Other Polygons
Middle School Algebra
Middle School Geometry
Middle School Puzzles
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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