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## Prime Factorization of Natural Numbers – Lucid Explanation of the Method of Finding Prime Factors

**Key factors (PFs): **

The factors of a natural number that are prime numbers are called the PFs of that natural number.

*Examples : *

The 8 elements are 1, 2, 4, 8.

Of these, only two are PF.

Also 8 = 2 x 2 x 2;

The 12 factors are 1, 2, 3, 4, 6, 12.

Of these, only 2, 3 are PFs

Also 12 = 2 x 2 x 3;

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.

Of these, only 2, 3,5 are PFs

Again 30 = 2 x 3 x 5;

The 42 factors are 1, 2, 3, 6, 7, 14, 21, 42.

Of these, only 2, 3, 7 are PFs

Again 42 = 2 x 3 x 7;

In all the examples here, each number is expressed as a product of PFs

In fact, we can do that for any natural number ( ≠ 1).

**Maximum PFs : **

In a PF ‘p’ of a natural number ‘n’, the multiple of ‘p’ is the greatest negative of ‘a’ where ‘p^a’ divides ‘n’ exactly.

*Examples : *

We have 8 = 2 x 2 x 2 = 2^3.

2 is the 8th PF.

Multiplying 2 is 3.

Again, 12 = 2 x 2 x 3 = 2^2 x 3

2 and 3 are 12 PFs.

The multiple of 2 is 2, and the multiple of 3 is 1.

**Prime Factorization : **

Expressing a given natural number as a product of PFs is called Prime Factorization.

or Prime Factorization is the process of finding all PFs, and their maximum for a given natural number.

The Prime Factorization of a Natural Number is unique without order.

This statement is called the Fundamental Theorem of Arithmetic.

**Prime Factorization method for a given natural number :**

**STEP 1 : **

Divide the given natural number by the smallest PF

**STEP 2 : **

Divide the quotient obtained in step 1, by its smallest PF.

Continue dividing each of the following quotients by their smallest PF, until the final quotient is 1.

**STEP 3 : **

Express the given natural number as the product of all these conditions.

This becomes the Prime Factorization of a natural number.

The steps and method of presentation will be clear with the following examples.

**Solved Example 1 :**

Find the Prime Factorization of 144.

*Solution: *

2 | 144

———–

2 | 72

———–

2 | 36

———–

2 | 18

———–

3| 9

———–

3| 3

———–

the end| 1

See the demonstration method given above.

144 is divided by 2 to get a quotient of 72 which is again

divided by 2 to get the quotient of 36 which is again

divided by 2 to get a quotient of 18 which is again

divided by 2 to get a quotient of 9 which is again

divided by 3 to get a quotient of 3 which is again

divide by 3 to get a quotient of 1.

See how the PFs are presented to the left of the vertical line

and quotients to the right, below the horizontal line.

Now 144 will be expressed as the product of all PFs

which are 2, 2, 2, 2, 3, 3.

So, Prime Factorization of 144

= 2 x 2 x 2 x 2 x 3 x 3. = 2^4 x 3^2 Ans.

**Solved Example 2 : **

Find the Prime Factorization of 420.

*Solution: *

2 | 420

———–

2 | 210

———–

3| 105

———–

5| 35

———–

7| 7

———–

the end| 1

See the demonstration method given above.

420 is divided by 2 to get the quotient of 210 which is again

divided by 2 to get the quotient of 105 which is again

divided by 3 to get the quotient of 35 which is again

divide by 5 to get a quotient of 7 which is again

divide by 7 to get a quotient of 1.

See how the PFs are presented to the left of the vertical line

and quotients to the right, below the horizontal line.

Now 420 will be expressed as the product of all PFs

which are 2, 2, 3, 5, 7.

So, the Prime Factorization of 420

= 2 x 2 x 3 x 5 x 7 = 2^2 x 3 x 5 x 7. Ans.

Sometimes you may need to use the Divisibilty Rules to find the smallest PF we need to divide by.

Let’s see an example.

**Solved Example 3 : **

Find the Prime Factorization of 17017.

*Solution : *

Given number = 17017.

Obviously this is not divisible by 2. (the last number is not exact).

Sum of numbers = 1 + 7 + 0 + 1 + 7 = 16 is not divisible by 3

and then the given number is not divisible by 3.

Since the last digit is neither 0 nor 5, it is divisible by 5.

Let’s use the division rule of 7.

Twice the last number = 2 x 7 = 14; number remaining = 1701;

difference = 1701 – 14 = 1687.

Twice the last digit 1687 = 2 x 7 = 14; number remaining = 168;

difference = 168 – 14 = 154.

Twice the last digit 154 = 2 x 4 = 8; number remaining = 15;

Difference = 15 – 8 = 7 is divisible by 7.

So, the given number is divisible by 7.

Let’s divide by 7.

17017 ÷ 7 = 2431.

As division by 2, 3, 5 is ruled out,

division by 4, 6, 8, 9, 10 is also excluded.

Let’s use the rule of dividing by 11.

Sum of 2431 bits = 2 + 3 = 5.

The sum of the remaining numbers is 2431 = 4 + 1 = 5.

Difference = 5 – 5 = 0.

So, 2431 is divisible by 11.

2431 ÷ 11 = 221.

As division by 2 is excluded, division by 12 is also excluded.

Let’s use the division rule of 13.

Four times the last digit of 221 = 4 x 1 = 4; number remaining = 22;

sum = 22 + 4 = 26 divided by 13.

So, 221 is divisible by 13.

221 ÷ 13 = 17.

Let’s talk about all these categories below.

7| 17017

———–

11 | 2431

———–

13 | 221

———–

17 | 17

———–

the end| 1

Thus, the Prime Factorization of 17017

= 7 x 11 x 13 x 17. Ans.

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