### Prime Factorization:

The process of expressing a given number as a product of prime factors is called a **Prime Factorization** or **Complete Factorization** of the given number.

**OR**

**Prime Factorization** is finding which prime numbers multiply together to make the original number.

If a given number is a composite number, it can be written as the product of two of its factors. These factors in turn are also either prime or composite. If composite, the factors can be split up further. Thus, we can express any number as the product of simpler and simpler factors until all the factors are prime numbers.

**Example:** Prime Factorization of 120.

We observe that in each case, 120 is expressed as a product of prime factors consisting of only 2, 3 and 5. Therefore, we say that 120 has been expressed as a product of **prime factors**.

Such factorizations are called prime factorizations.

Methods of Prime factorization:

1) **Division method for prime factorization:**

To find prime factorization of a number by division method,

Step 1) Start dividing the number by the least prime factor.

Step 2) Continue division till the resulting number to be divided is 1.

**Example:** Prime Factorization of 210.

Prime factorization of 210 = 7 x 3 x 5 x 2

Prime factorization of 38 are 2 x 19.

2) **Factor Tree method:**

To find prime factorization of 60 using factor tree method, we proceed as follow:

Step 1) Express 60 as a product of two numbers.

Step 2) Factorize 4 and 15 further, since they are composite numbers.

Step 3) Continue till all the factors are prime numbers.

Prime factorization of 60 = 2 x 2 x 3 x 5.

Prime actorization of 38 are 2 x 19.

#### Prime Factorization Property or Fundamental Theorem of Arithmetic:

Every composite number can be factorized into prime factors in one and only one way, except, for the order of the factors.

It follows from the above property, that either a number is prime or it can be uniquely expressed as the product of prime factors.

To find the unique prime factorization of a number n, we may use the following steps:

STEP 1) Choose the smallest prime p which divides n. Divide n by p. Let the quotient be m1.

Then, n =p x m1.

STEP 2) If m1 is prime, then p x m1 is the prime factorization of n. Otherwise, we choose the smallest prime q which divides m1. Let the quotient be m2.

Then, n =(p x q x m2).

If m2 is prime, then (p x q x m2) is the prime factorization of n. Otherwise we continue the same process till we get a prime quotient.

**Example:** Prime factorization of 72.

In the prime factorization of 72, we observe that there are several prime factorizations of 72. But, in each prime factorization prime factors are same (2, 2, 2, 3, 3) their order may differ. Thus the prime factorization of 72 is unique. This is not only true for 72 but it holds true for every composite number.

Once we know the prime factors of a number, we can determine ALL the factors of that number, even large numbers. The factors can be found by building all the possible products of the prime Factors.

**Prime Factorization can be used in other situations as follows:**

(1) Determining whether one number is divisible by another number

(2) Determining the greatest common factor (HCF) of two numbers

(3) Reducing fractions .

(4) Finding the least common multiple (LCM) of two (or more) numbers

(5) Simplifying square roots

(6) Determining the exponent on one side of an equation with integer constraints