Factors and Rectangles

Date: 06/25/98 at 18:22:52
From: amy
Subject: Factors and rectangles

I'm trying to figure out what a rectangle and factors have in common.

Date: 06/26/98 at 12:48:50
From: Doctor Peterson
Subject: Re: Factors and rectangles

Hi, Amy - 

The answer is pretty simple, but it can give you a lot to think about.

Since the area of a rectangle is the product of the lengths of the 
sides, the sides are always factors of the area.

It's easier to picture if you think of building a rectangle out of 
objects - maybe little square blocks, or maybe just arranging coins in 
a rectangle. If I gave you a handful (or a roomful) of blocks and 
asked you to make a rectangle out of them, you would have to decide 
what size rectangle you should build. Not all rectangles would work. 
For example, if you had 14 blocks and tried building a rectangle 5 
blocks across, you would find that you didn't have enough to make the 
third row:

    O O O O O
    O O O O O
    O O O O

That's because 5 is not a factor of 14. If you factor 14, you will 
find that 14 = 2 * 7, so the only rectangles you could make would be 
2 by 7 or 7 by 2:

    O O O O O O O    O O
    O O O O O O O    O O
                     O O
                     O O
                     O O
                     O O
                     O O

Oops - I forgot you could also make a 1 by 14 or a 14 by 1 rectangle -
they're easy to forget!

The fun part comes when you have a number that can factor in more 
ways. For example, with 36 blocks you could make all these rectangles:

    1 * 36
    2 * 18
    3 * 12
    4 *  9
    6 *  6
    9 *  4
   12 *  3
   18 *  2
   36 *  1

Try playing with rectangles for a while. It can really help you get a 
feel for how multiplication and factoring work!

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