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,...