# Math Basics — Least Common Multiple

If GRE math concepts were people, **Least Common Multiple** (LCM) would hardly be the most noticeable person in the room. After all, we have the really big personalities who always elbow for attention: Probability, Work Rates and Combinations (none of whom would win the popularity context). Then, there are Exponents standing together menacingly with Coordinate Geometry, with Scientific Notation sneering up at us. Finally, squished in the corner, making room for all these other concepts, is our little wallflower: Least Common Multiple. So self-effacing—introducing itself with a Least—and meek, this concept is more familiar in its diminutive: LCM.

But let’s not forget that finding the Least Common Multiple can help us on questions ranging from work rates to number properties. So, now is our cue to walk up to meek LCM and say hello.

First off, finding LCM for a group of numbers requires three steps.

## 1. Find the Factors that the Numbers Have in Common.

Take a look at the following numbers: 14, 35 and 70.

Which factors do they have in common? To answer this question, we need to break down each number into its prime factors.

14 = 7 x 2

35 = 5 x 7

70 = 5 x 7 x 2

All three of them have the number 7. Also, 35 and 70 have 5 in common, and 14 and 70 can both be divided by 2. Therefore, the shared factors are 7, 5, and 2. Note that we include each shared factor only once, even if, as is the case for the number 7, it shows up in all three numbers.

## 2. Find the Lone Factors.

Take a look at the following numbers: 15, 8, and 12.

Let’s break down each to prime factors:

15 = 3 x 5

8 = 2 x 2 x 2

12 = 2 x 2 x 3

Is there any factor that is not shared by the other two factors? Most saliently, there is the one 5. Are there any other lone factors? Actually, there is one 2, all by itself. Therefore, the lone factors are 5 and 2. Notice that two of the 2s overlap for both 8 and 12. But the number 8 has three factors of 2. Therefore, that extra two is a lone factor.

Let’s now find the shared factors (step 1) of 15, 8, and 12, so we can go on to the third step. We can see that the shared factors are 2, 2, and 3.

## 3. Multiply the Product of the Shared Factors by the Product of the Lone Factors

Shared Factors : 2 x 2 x 3 = 12

Lone Factors: 2 x 5 = 10

12 x 10 = 120.

Therefore the LCM is 120.

And just like that we’ve met, not the life of the party, but an indispensable concept to doing well on the GRE math section.