# GCF of 96 and 108

By the end of this lesson, you will be able to

- describe the greatest common factor of 96 and 108.
- identify the factor of 96 and 108.
- find out the GCF of 96 and 108.

## Greatest Common Factor of 96 and 108

If one asks, what is the GCF of 96 and 108? The clear answer in a word, the greatest common factor of 96 and 108 is 12. Note that GCF is the short form of greatest common factor (GCF).

There are number of ways to determine the highest common factor of some numbers.

### Method 1

Let's go to find out the highest common factor of 96 and 108 by applying prime factorization method.

Step 1

Identify all the possible prime factors of 96.

Begin, by dividing 96 by first prime number 2. And continue this process i.e. continue dividing by 2 until getting a remainder other than zero. Again divide the number by next prime numbers 3, 5, 7, 11, 13, ... until the number left takes the form of a prime number.

96 ÷ 2 = 48

∴ 96 = 2 × 48

Again, 48 ÷ 2 = 24

∴ 96 = 2 × 2 × 24

Again, 24 ÷ 2 = 12

∴ 96 = 2 × 2 × 2 × 12

Again, 12 ÷ 2 = 6

∴ 96 = 2 × 2 × 2 × 2 × 6

And lastly, 6 ÷ 2 = 3

∴ 96 = 2 × 2 × 2 × 2 × 2 × 3

Therefore, 2, 2, 2, 2, 2, and 3 are the prime factors of 96.

Step 2

Again, find all the possible prime factors of 108.

108 ÷ 2 = 54

∴ 108 = 2 × 54

Again, 54 ÷ 2 = 27

∴ 108 = 2 × 2 × 27

Again, 27 ÷ 3 = 9

∴ 108 = 2 × 2 × 3 × 9

And lastly, 9 ÷ 3 = 3

∴ 108 = 2 × 2 × 3 × 3 × 3

Therefore, 2, 2, 3, 3, and 3 are the prime factors of 108.

Step3

Arrange the prime factors of 96 and 108 together.

∴ 96 = 2 × 2 × 2 × 2 × 2 × 3

∴ 108 = 2 × 2 × 3 × 3 × 3

The prime factors of 96 are 2, 2, 2, 2, 2, and 3.

And the prime factors of 108 are 2, 2, 3, 3 and 3.

Step 4

The multiplication of all common prime divisors is the HCF of 96 and 108.

It is seen, 2, 2 and 3 are the common prime divisor.

Hence, the greatest common divisor of 96 and 108 = 2 × 2 × 3.

∴ GCF of 96 and 108 = 12.

Note that if there is no common prime factor among numbers, the hightest common factor is 1.

### Method 2

Step 1

Find all factors of 96.

96 ÷ 1 = 96

∴ 96 = 1 × 96

Again, 96 ÷ 2 = 48

∴ 96 = 2 × 48

Again, 96 ÷ 3 = 32

∴ 96 = 3 × 32

Again, 96 ÷ 4 = 24

∴ 96 = 4 × 24

Again, 96 ÷ 6 = 16

∴ 96 = 6 × 16

Again, 96 ÷ 8 = 12

∴ 96 = 8 × 12

Arrange those factors together.

96 = 1 × 96

96 = 2 × 48

96 = 3 × 32

96 = 4 × 24

96 = 6 × 16

96 = 8 × 12

So, all the factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96.

Step 2

Identify all factors of 108.

Find all factors of 108.

108 ÷ 1 = 108

∴ 108 = 1 × 108

Again, 108 ÷ 2 = 54

∴ 108 = 2 × 54

Again, 108 ÷ 3 = 36

∴ 108 = 3 × 36

Again, 108 ÷ 4 = 27

∴ 108 = 4 × 27

Again, 108 ÷ 6 = 18

∴ 108 = 6 × 18

Again, 108 ÷ 9 = 12

∴ 108 = 9 × 12

Arrange those factors together.

108 = 1 × 108

108 = 2 × 54

108 = 3 × 36

108 = 4 × 27

108 = 6 × 18

108 = 9 × 12

So, all the factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108.

Step3

Arrange all the factors of 96 and 108 together in two seperate rows.

All the factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96.

All the factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108.

Step 4

Identify the greatest common factor of 96 and 108.

It is seen, the highest common factor of 96 and 108 is 12.

Therefore, the GCF of 96 and 108 is 12.

### Method 3

If the product of 96 and 108 is divided by the least common multiple (LCM) of 96 and 108, the resultant value is the greatest common divisor of 96 and 108.

Thus, GCF of 96 and 108 i.e.

GCF(96,108) = Product of 96 and 108LCM of 96 and 108

or, GCF(96,108) = 96 × 108LCM(96,108)

or, GCF(96,108) = 10368864

or, GCF(96,108) = 864 × 12864

∴ GCF(96,108) = 12

Therefore, the GCF of 96 and 108 is 12

### Frequently Asked Questions (FAQ)

FAQ on GCF of 96 and 108

Answer: The GCF of 96 and 108 is 12.

Answer: All possible factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. So, there are twelve positive integers those are all possible factors of 96. The smallest factor of 96 is 1. And the greatest factor of 96 is 96.

Answer: All possible factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108. So, there are twelve positive integers those are all possible factors of 108. The smallest factor of 108 is 1. And the greatest factor of 108 is 108.

Answer: 96 = 2 × 2 × 2 × 2 × 2 × 3. Therefore, all prime factors of 96 are 2 and 3. The smallest prime factor of 96 is 2. And the greatest prime factor of 96 is 3.

Answer: 108 = 2 × 2 × 3 × 3 × 3. Therefore, all prime factors of 96 are 2 and 3. The smallest prime factor of 108 is 2. And the greatest prime factor of 108 is 3.

Answer: Step 1: Identify the prime factors of 96.

Now, 96 = 2 × 2 × 2 × 2 × 2 × 3.

Step 2: Identify the prime factors of 96.

Now, 108 = 2 × 2 × 3 × 3 × 3.

Step 3: Gather all prime factors those are common in step 1 and step 2. And they are 2, 2 and 3.

Step 4: Multiply all prime factors collected in step 3.

And it is 2 × 2 × 3 = 12

Therefore, the GCF of 96 and 108 is 12.