# GCF AND LCM BY PRIME FACTORIZATION

**I. To Find the GCF**

**GCF** stands for Greatest Common Factor. Factors are numbers multiplied together. Sometimes, it’s necessary to know the GCF of 2 or more numbers. Why? I’m glad you asked. Basically, the GCF helps to efficiently and quickly reduce fractions. One way to determine the GCF is simply by prime factorization.

**When to use :** to reduce fractions

**Skills needed: **multiplication facts, prime factorization

**Example: 12 & 36 Find the GCF**

**Step 1:** Find the Prime factors for

12: 2 x 2 x 3

36: 2 x 2 x 3 x 3

**Step 2:** Multiply each factor that has a match

GCF = 2 x 2 x 3 = 12

**Example: Find the GCF for 48 and 72**

**Step 1:** Find the prime factors for

48: 2 x 2 x 2 x 2 x 3

72: 2 x 2 x 2 x 3 x 3

**Step 2: **Multiply each factor that has a match

GCF = 2 x 2 x 2 x 3 = 24

**Example: Find the GCF for 56, 70, and 98**

**Step 1: **Find the prime factors for

56: 2 x 2 x 2 x 7

70: 2 x 5 x 7

98: 2 x 7 x 7

**Step 2:** Multiply each factor that has a match in all 3 numbers

GCF = 2 x 7 = 14

**Application**

Reduce or simplify the fraction, 48/72.

1. Find the GCF (We know that the GCF for 48 & 72 is 24)

2. Divide the numerator and denominator by 24. (48/24 and 72/24)

3. Thus 48/72 = 2/3

**II. To Find the LCM**

**When to use:**when adding or subtracting unlike denominators, the LCM is needed to find the least common denominator

**Skills needed: **multiplication facts, prime factorization

LCM stands for Least Common Multiple (LCM). - Multiples are the result of repeated addition or what you referred to as skip counting in Elementary school

For example, the multiples of 2 are below

2 >>> 2, 4, 6, 8, 12, 14, 16, 18, 24 . . .

In the above section, we found the GCF. Let’s use the first two examples, but find the LCM.

**Example: Find the LCM of 12 and 36**

**Step 1:** Find the Prime factors for

12 = 2 x 2 x 3

36 = 2 x 2 x 3 x 3

**Step 2:** Multiply each prime factor the greatest number of times it appears in any one factorization.

The most number of times 2 appears in either factorization is twice. The most number of times 3 appears in either factorization is twice. Thus,

LCM = 2 x 2 x 3 x 3 = 36

**Example: Find the LCM for 48 and 72**

**Step 1:** Find the prime factors for

48: 2 x 2 x 2 x 2 x 3

72: 2 x 2 x 2 x 3 x 3

**Step 2:** Multiply each prime factor the greatest number of times it appears in any one factorization. The most number of times 2 appears in either factorization is four times. The most number of times 3 appears in either factorization is twice. Thus,

LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144

**Example: Find the LCM for 15, 25, and 6**

**Step 1: **Find the prime factors for

6 = 2 x 3

15 = 3 x 5

25 = 5 x 5

**Step 2:** Multiply each prime factor the greatest number of times it appears in any one factorization. The most number of times 2 appears in either factorization is once. The most number of times 3 appears in either factorization is once. The most number of times 5 appears in either factorization is twice. Thus,

LCM = 2 x 3 x 5 x 5 = 150

**Application:**

**Add 5/12 and 7/36**

The denominators are not the same. So, find the LCM, also referred to as the least common denominator (LCD). The LCD is 36. (Refer to example above)

The equivalent fraction for 5/12 using 36 as the denominator is 15/36. 7/36 stays the same. Therefore 15/36 + 7/36 = 22/36.

Reduce or simplify 22/36 to its lowest terms. Use the greatest common factor (GCF).

The GCF for 22 and 36 is 2.

Divide both 22 and 36 by 2. 22/36 = 11/18

**Summary: 5/12 + 7/36 = 15/36 + 7/36 = 22/36 = 11/18**