Form A Quadratic Formula With The Points With Linear Algebra 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 (PFs):

Factors of natural numbers which are prime numbers are called PF of natural numbers.

Examples:

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

Only 2 of them are PF.

Also 8 = 2 x 2 x 2;

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

Out of them only 2 thousand 3 people are PF

Also 12 = 2 x 2 x 3;

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

Out of them 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.

Out of them only 2, 3, 7 are PF

Also 42 = 2 x 3 x 7;

Here in all these examples, each number is expressed as a product of PFs

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

Multiplicity of PFs:

For a PF ‘p’ of a natural number ‘n’, the product 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.

A product of 2 is 3.

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

2 and 3 are PFs of 12.

A product of 2 is 2, and a product of 3 is 1.

Prime Factorization:

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

or prime factorization is the process of finding all PFs including their product for a given natural number.

A prime factorization for a natural number is unique except for order.

This statement is called the Fundamental Theorem of Arithmetic.

Prime factorization method of given natural numbers:

Step 1:

Divide the given natural number by its smallest PF

Step 2:

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

Divide each subsequent quotient by their smallest PFs until the last quotient is 1.

Step 3:

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

It is a prime factorization of natural numbers.

The steps and method of presentation will be clear through 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 presentation method given above.

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

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

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

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

Divide by 3 to get a quotient of 3 which repeats

Divide by 3 to get a quotient of 1.

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

And to the right, the quotient below the horizontal line.

Now 144 is to be expressed as the product of all PFs

Which is 2, 2, 2, 2, 3, 3.

So, the 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 presentation method given above.

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

Divide by 2 to get the quotient of 105 which again occurs

Divide 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.

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

And to the right, the quotient below the horizontal line.

Now 420 must be expressed as the product of all PFs

Which is 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. Answer.

Sometimes you may need to apply division rules to find the minimum PF we must divide by.

Let’s look at 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 digit.)

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

And therefore the given number is not divisible by 3.

Cannot be divided by 5 because the last digit is not 0 or 5.

Let’s apply the division rule of 7.

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

Difference = 1701 – 14 = 1687.

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

Difference = 168 – 14 = 154.

154 = 2 x 4 = twice the last digit of 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.

Division by 2, 3, 5 is rejected,

Divisibility by 4, 6, 8, 9, 10 is also denied.

Let’s apply the divisibility rule by 11.

2431 = Sum of alternate digits of 2 + 3 = 5.

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

Difference = 5 – 5 = 0.

So, 2431 is divisible by 11.

2431 ÷ 11 = 221.

Just as divisibility by 2 is denied, division by 12 is also denied.

Let’s apply the division rule of 13.

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

Sum = 22 + 4 = 26 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

———-

The end 1

 

Thus, the prime factorization of 17017

= 7 x 11 x 13 x 17. Answer.

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