- Using graphics, this method provides a visual alternative for computational methods.

The alternative method described here is graphic. It requires no factoring or division. In fact, this visual method requires no computation whatsoever. Instead you apply the graphic method for finding the GCF, do some additional drawing, count and finally multiply two numbers.

This graphic method
does not rely on knowing any common factors. It employs *simple division*
and each division step is progressively simpler. Or, you may even use this
method __without ever using division__.
At most, if you so choose (as stated above), you **only need*** subtraction.*

__This method has several
advantages__:

- It also provides a visual means to explain the concept of least common multiple (LCM) and why the various methods, including those that employ factoring, work.

- Being an alternative, it demonstrates that there are different ways to reach the same mathematical destination.

- This method demonstrates the close relationship between arithmatic and geometry.
- the GCF (GCD), by
- the GCF count along the short edge, by
- the GCF count along the long edge.

Since* *** 7**
and

we get their LCM by simply multiplying

Look closely at the
GCF (GCD) square, which you replicated along the edges of the original
rectangle. What is the size of the GCF square, i.e. the length of one side
of the GCF square? — As its name implies, the size of the GCF square is
the greatest common factor of the two numbers that are represented by sides
of the rectangle. In the first example the answer is ** 4**
and in the second it is

It is important to
keep in mind that the size of each edge consists of the multiplication
of the GCF by its GCF-square count. Therefore, when we multply the GCF-square
count along one edge by the length of the ** other** edge we exclude
from the multiplication one occurance of the GCF (GCD). Examining the resulting
product, we see that it always consists of the multiplication of these
three numbers

As we saw above,
the product of the first and the second is the length of the short edge
and the product of the first and the third is the length of the long edge.

This method provides a visual demonstration why the LCM equals the product of the two original numbers divided by their GCF (GCD).

Copyright © 1993-2008 Ten Ninety, All Rights Reserved

Last Update: Sep. 3, 2008