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

**Prime factors (PF): **

Factors of a natural number that are prime numbers are called PFs of that natural number.

*Examples: *

The factors of 8 are 1, 2, 4, 8.

Of these, only 2 are the PF.

Also 8 = 2 x 2 x 2;

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

Of these, only 2, 3 are the PF

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 PF

Also 30 = 2 x 3 x 5;

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

Of these, only 2, 3, 7 are PF

Also 42 = 2 x 3 x 7;

In all these examples here, each number is expressed as a product of PF

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

**Multiplicity of PF: **

For a PF ‘p’ of a natural number ‘n’, the multiplicity of ‘p’ is the largest exponent ‘a’ for which ‘p^a’ divides ‘n’ exactly.

*Examples: *

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

2 is the PF of 8.

The multiple of 2 is 3.

Also, 12 = 2 x 2 x 3 = 2^2 x 3

2 and 3 are the PFs of 12.

The multiplicity of 2 is 2 and the multiplicity of 3 is 1.

**Prime factorization: **

Expressing a given natural number as a product of PF is called prime factorization.

o Prime factorization is the process of finding all PFs, along with their multiplicity, for a given natural number.

The prime factorization for a natural number is unique except for the order.

This statement is called the Fundamental Theorem of Arithmetic.

**Method of prime factorization of a given natural number:**

**STEP 1 : **

Divide the given natural number by its smallest PF

**STEP 2: **

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

Continue dividing each subsequent quotient by its smallest PF, until the last quotient is 1.

**STEP 3: **

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

This becomes the prime factorization of the natural number.

The steps and method of presentation will be clear from 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

———-

end| 1

See submission method above.

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

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

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

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

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

divided by 3 to get the quotient of 1.

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

and the quotients on the right, below the horizontal line.

Now 144 must 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.

**Example 2 solved: **

Find the prime factorization of 420.

*Solution: *

2 | 420

———-

2 | 210

———-

3| 105

———-

5| 35

———-

7| 7

———-

end| 1

See submission method above.

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

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

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

divided by 5 to get the quotient of 7 which again is

divided by 7 to get the quotient of 1.

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

and the quotients on the right, below the horizontal line.

Now 420 must be expressed as the product of all PFs

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

So 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 apply the Divisibility Rules to find out the minimum PF with which we need to perform the division.

Let’s see an example.

**Example 3 solved: **

Find the prime factorization of 17017.

*Solution : *

The given number = 17017.

Obviously, this is not divisible by 2 (the last digit is not even).

The sum of the digits = 1 + 7 + 0 + 1 + 7 = 16 is not divisible by 3

and therefore the given number is not divisible by 3.

Since the last digit is neither 0 nor 5, it is not divisible by 5.

We apply the rule of divisibility of 7.

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

difference = 1701 – 14 = 1687.

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

difference = 168 – 14 = 154.

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

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

Hence the given number is divisible by 7.

We divide by 7.

17017 ÷ 7 = 2431.

Since divisibility by 2, 3, 5 is ruled out,

Divisibility by 4, 6, 8, 9, 10 is also ruled out.

We apply the rule of divisibility by 11.

Alternate digit sum of 2431 = 2 + 3 = 5.

Sum of the remaining digits of 2431 = 4 + 1 = 5.

Difference = 5 – 5 = 0.

So 2431 is divisible by 11.

2431 ÷ 11 = 221.

Since divisibility by 2 is ruled out, divisibility by 12 is also ruled out.

We apply the rule of divisibility of 13.

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

sum = 22 + 4 = 26 is divisible by 13.

So 221 is divisible by 13.

221 ÷ 13 = 17.

We present all these divisions below.

7| 17017

———-

11 | 2431

———-

13 | 221

———-

17 | 17

———-

end| 1

Thus, the prime factorization of 17017

= 7 x 11 x 13 x 17. Ans.

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