# Least Common Multiple Example

There are two ways to find the LCM given in the textbook (Basic Mathematical Skills with Geometry, 8th Ed. by Baratto and Bergman)

Let’s look at example #4 on page 192 in a little more detail.

One way of doing this problem would be the ‘Listing Method’.  So, to find the LCM of 10 and 18, we could do the following:

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, etc…

18, 36, 54, 72, 90, 108, 126, 144, 162, 180, etc…

Note that 90 is the LCM, but 180 is also in common, it’s just not the lowest number in common.  However, we could have already guessed that 180 is in common since 10 times 18 = 180.

Let’s look at the ‘Prime Factorization Method’ now.

10 = 2 x 5  and 18 = 2 x 3 x 3

We have to line up the factors vertically.  This means that a 2 can only be lined up over a 2, a 3 can only be lined up over a 3, a 5 can only be lined up over a 5, etc.

10 = 2           x 5 (The 5 moved over because it can’t be over a 3)

18 = 2 x 3 x 3

Now we count one number from each column.  See the bolded numbers above.  Note that one of the 2’s was repeated (since it was stacked on top of another 2).  If there numbers stacked on top of each number like that, we only count it once.

Thus, we have 2 x 3 x 3 x 5 = 90.

Now, remember the listing method?  Remember how 90 and 180 were both in common?  180 is actually too high simply because it counts the ‘extra 2′ that was stacked when it didn’t need to be counted.