Prime factorization is a way of expressing a number as a product of prime numbers. Due to the commutative property of multiplication the order of numbers in a product does not matter. Hence the prime factorization of a number is unique.

The Prime factorization of the number 56 is , 56 = 2 x 2 x 2 x 7 or using exponents 56 = 2

**Example:**The Prime factorization of the number 56 is , 56 = 2 x 2 x 2 x 7 or using exponents 56 = 2

^{3}x 7.## How to find the prime factorization ?

**Example**:

Find the prime factorization of each number 156 and 255.

First let us find the prime factorization of 156.

Since 156 is an even number, it is divisible by 2. Division of 156 by 2 yields a quotient 78. The number can be now written in factored form as,

156 = 2 x 78

Again 78 is divisible by 2 and the division yields the quotient 39. Now the factored form of 156 is

156 = 2 x 2 x 39.

The sum of the digits of the number 39 is 3+9 = 12 which is divisible by 3. Hence 39 is divisible by 3. The division of 39 by 3 yields the quotient 13. The quotient 13 is a prime number. Hence the division cannot be continued. The required prime factorization is therefore,

**156 = 2 x 2 x 3 x 13**or using exponents

**156 = 2**

^{2}x 3 x 13.A factor tree gives a visual explanation of the process involved.

156

/\

/ \

Prime factor →

**2**78

/\

/ \

Prime factor →

**2**39

/\

/ \i

Prime factor →

**3**

**13**← Prime factor.

Now to find the prime factorization of 255.

Since the ones digit is 5, the number is divisible by 5. The quotient on division of 255 by 5 is 51.

255 = 5 x 51

The sum of the digits of the number 51, 5+1 =6 is divisible by 3. Hence 51 is divisible by 3. The division of 3 yields a quotient 17.

51 = 3 x 17

and 255 = 5 x 3 x 17. Since the last quotient is a prime number, the division is not continued. Thus the required prime factorization is

**255 = 3 x 5 x 17.**

## Prime factorization to find the GCF of numbers

**Example:**

Find the GCF of the numbers 36 and 48.

First we need to write the prime factorizations for the numbers given.

36 = 2 x 2 x 3 x 3 and 48 = 2 x 2 x 2 x 2 x 3.

Picking the highlighted common factors and multiplying them,

**GCF of 36 and 48 = 2 x 2 x 3 = 12**.

## Finding LCM using prime factorization

One method of finding the LCM is to list the multiples of the numbers given and picking the first occurring common multiple in the lists as the LCM. This will work fine for smaller numbers. But for larger numbers, finding the multiples is not an easy task. So we can follow the following steps.

**Write the prime factorization of the given numbers.****Find the GCF of the numbers using the prime factorization.****The LCM is the product of GCF and the left out non common numbers in the prime factorization.**

**Example:**

Find the LCM of the numbers 36 and 48 using prime factorization.

The prime factorization for 36 and 48 are

36 = 2 x 2 **x 3** **x 3** and 48 = **2 x 2**** x** **2 x 2 x 3**

Picking the common factors, the GCF = **2 x 2 x 3 = 12**

Now LCM is got by multiplying the GCF with the left out factors in the prime factorization.

LCM of 36 and 48 = **12**** x**** 3 x 2 x 2** = **144.**