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

Key Factors (PF):

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

Example:

Factors of 8 are 1, 2, 4, 8.

Of these, only 2 are PF.

Also 8 = 2 x 2 x 2;

Factors of 12 are 1, 2, 3, 4, 6, 12.

Of these, only 2, 3 are PF

Also 12 = 2 x 2 x 3;

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 of these examples here, each number is expressed as a product of PFs

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

Multiplicity of PF:

For a PF ‘p’ of a natural number ‘n’, the multiple of ‘p’ is the largest index of ‘a’ that divides ‘p^a’ exactly ‘n’.

Example:

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 PF of 12.

A multiple of 2 is 2 and a multiple of 3 is 1.

First Factorization:

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

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

The prime factorization for a Natural Number is unique except for the sequence.

This statement is called the Fundamental Theorem of Arithmetic.

Prime Factorization Method of a given natural number:

STEP 1 :

Divide the given natural number by its smallest PF

STEP 2:

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

Continue, dividing each of the last numbers by their smallest PFs, until the last number is 1.

STEP 3:

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

This becomes the Prime Factorization of a natural number.

The following examples will illustrate the steps and method of presentation.

Solved Example 1:

Find the prime factorization of 144.

Solution:

2 | 144

———-

2 | 72

———-

2 | 36

———-

2 | 18

———-

3 | 9

———-

3 | 3

———-

last | 1

See the presentation method given above.

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

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

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

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

is divided by 3 to get the number 3 which is again

divided by 3 to get a ratio of 1.

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

and quotients on the right, below the horizontal line.

Now 144 is defined as the product of all PFs

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

Hence, Prime Factorization of 144

= 2 x 2 x 2 x 2 x 3 x 3. = 2^4 x 3^2 Answer.

Solved Example 2:

Find the prime factorization of 420.

Solution:

2 | 420

———-

2 | 210

———-

3 | 105

———-

5 | 35

———-

7 | 7

———-

last | 1

 

See the presentation method given above.

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

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

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

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

divided by 7 to get a ratio of 1.

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

and quotients on the right, below the horizontal line.

Now 420 is defined as the product of all PFs

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

Hence, Prime Factorization of 420

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

Sometimes you may have to apply the Dividend Rules to find the lowest PF with which to divide.

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 (not even the last number).

Sum of numbers = 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 not 0 or 5, it is not divisible by 5.

Let’s apply the division rule of 7.

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

difference = 1701 – 14 = 1687.

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

difference = 168 – 14 = 154.

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

difference = 15 – 8 = 7 divided by 7.

Therefore, the given number is divisible by 7.

Let’s divide by 7.

17017 ÷ 7 = 2431.

Since division by 2, 3, 5 is rejected,

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

Let’s apply the rule of division by 11.

The sum of the alternate numbers of 2431 = 2 + 3 = 5.

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

Difference = 5 – 5 = 0.

So, 2431 is divisible by 11.

2431 ÷ 11 = 221.

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

Let’s apply the division rule of 13.

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

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

So, 221 is divisible by 13.

221 ÷ 13 = 17.

Let us present all these divisions below.

7 | 17017

———-

11 | 2431

———-

13 | 221

———-

17 | 17

———-

last | 1

 

Thus, the Prime Factorization of 17017

= 7 x 11 x 13 x 17. Answer.

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