Least Common Multiple (LCM) & Greatest Common Factor (GCF)

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF for short, of a list of numbers is the largest factor that divides into all of the numbers evenly.

Example: The GCF of 20 and 30 is 10 since 10 divides into 20 evenly 2 times and 10 divides into 30 evenly 3 times.

Where do we use the GCF? The GCF is used when we reduce a fraction to lowest terms. For example, we reduce the fraction 20/30 by dividing out the GCF of 10 from both numerator and denominator, as shown below. The result is a reduced fraction of 2/3.

The quickest way to reduce a fraction is to divide out by the GCF!

Method 1 for finding GCF of a list of numbers: List all the possible factors of the numbers. Choose the largest common factor from the lists.

Example: Find the GCF of 24 and 40

Factors of 24 are all the numbers that divide in evenly. We can find these by listing out all the multiplied products that make 24.

24 = 1 x 24

24 = 2 x 12

24 = 3 x 8

24 = 4 x 6

24 = 6 x 4

24 = 8 x 3

24 = 12 x 2 Notice that we could have stopped at 4x6 since we just repeated what we had in reverse order. The factors are {1,2,3,4,6,8,12,24}

In a similar manner 40 = 1 x 40 = 2 x 20 = 4 x 10 = 5 x 8

The factors of 40 are {1, 2,4,5,8,10, 20}

The largest number common to both lists is 8, so the GCF of 24 and 40 is 8.

Application: The fraction 24/40 reduces to 3/5 when we divide 24 by 8 & 40 by 8.

Example: Find the GCF of 13 and 20

The factors of 13 are {1, 13} since 13. The factors of 20 are {1,2,4,5,10, 20}

The largest number common to both lists is 1 so the GCF is 1.

Applied to fractions, 13/20 would not be able to be reduced !

Method 2 for finding GCF of a list of numbers: Write the prime factorizations of the numbers. The GCF will be product of common factors present in all the prime factorizations.

Example: Find the GCF of 44 and 80.

44 = 2  22 = 2  2 11

80 = 2  40 = 2  2  20 = 2  2  2  10 = 2  2 2  2  5

There are two factors of 2 in both lists, so the GCF is 2  2 = 4

Example: Find the GCF of 24 and 48

24 = 2  12 = 2  2  6 = 2  2  2 3

48 = 2  24 = 2  2  12 = 2  2  2  6 = 2  2  2 2 3

There are three factors of 2 and one factor of 3 in both lists.

The GCF will be 2  2  2 3 = 24

Application of this: 24/48 reduces to ½ since we can divide the GCF of 24 out of both the numerator and denominator.

Why Method 2 For Finding the GCF?

Many students find this second method for finding the GCF a bit confusing. But we absolutely need this method for reducing an algebraic fraction like the one below that you will see in an Algebra II class. In the problem below, we would reduce the fraction by writing the top and bottom in factored form and then start canceling out factors common to both top and bottom

In Perspective:

If you are not going to take any more algebra classes ever, then Method 1 for both LCM and GCF will probably be sufficient for what you need. However, if you plan to take more algebra, particularly Algebra II, you should learn Method 2.