Prime factors (PF):
The factors of a natural number that are prime numbers are called FP of that natural number.
Examples:
The factors of 8 are 1, 2, 4, 8.
Of these, only 2 is the PF.
Also 8 = 2x2x2;
The factors of 12 are 1, 2, 3, 4, 6, 12.
Of these, only 2, 3 are the PF
Also 12 = 2x2x3;
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
Of these, only 2, 3.5 are the PF
Also 30 = 2x3x5;
The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
Of these, only 2, 3, 7 are the PF
Also 42 = 2x3x7;
In all these examples here, each number is expressed as a product of FP
In fact, we can do that for any natural number ( ≠ 1).
Multiplicity of FP:
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 = 2x2x2 = 2^3.
2 is the PF of 8.
The multiplicity of 2 is 3.
Also, 12 = 2 x 2 x 3 = 2^2 x 3
2 and 3 are the FP of 12.
The multiplicity of 2 is 2 and the multiplicity of 3 is 1.
Prime factorization:
Expressing a given natural number as the product of FP is called prime factorization.
o Prime factorization is the process of finding all the FPs, along with their multiplicity for a given natural number.
The prime factorization of a natural number is unique except for the order.
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 quotient obtained in step 1, by its smallest FP.
Continue dividing each of the subsequent quotients by its smallest FP, 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 the method of presentation will become clear with the following examples.
Worked example 1:
Find the prime factorization of 144.
Solution:
2 | 144
———-
2 | 72
———-
2 | 36
———-
2 | 18
———-
3 | 9
———-
3 | 3
———-
end| 1
See method of presentation given 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
Divide by 3 to get the quotient of 1.
See how the FPs are presented to the left of the vertical line
and the ratios to the right, below the horizontal line.
Now 144 must be expressed as the product of all the FPs
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 Years.
Worked example 2:
Find the prime factorization of 420.
Solution:
2 | 420
———-
2 | 210
———-
3 | 105
———-
5 | 35
———-
7 | 7
———-
end| 1
See method of presentation given 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
Divide by 7 to get the quotient of 1.
See how the FPs are presented to the left of the vertical line
and the ratios to the right, below the horizontal line.
Now 420 must be expressed as the product of all the 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. Years.
Sometimes it may be necessary to apply Divisibility Rules to know the minimum FP with which we have to carry out the division.
Let’s see an example.
Worked example 3:
Find the prime factorization of 17017.
Solution :
The given number = 17017.
Obviously this is not divisible by 2 (the last digit is not even).
Sum of 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 not 0 or 5, it is not divisible by 5.
Let’s apply the divisibility rule of 7.
Twice the last digit = 2 x 7 = 14; remaining number = 1701;
difference = 1701 – 14 = 1687.
Twice the last digit of 1687 = 2 x 7 = 14; remaining number = 168;
difference = 168 – 14 = 154.
Twice the last digit of 154 = 2 x 4 = 8; remaining number = 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.
Since divisibility by 2, 3, 5 is ruled out,
divisibility by 4, 6, 8, 9, 10 is also ruled out.
Let’s 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.
Let’s apply the divisibility rule of 13.
Four times the last digit of 221 = 4 x 1 = 4; remaining number = 22;
sum = 22 + 4 = 26 is divisible by 13.
So 221 is divisible by 13.
221 ÷ 13 = 17.
Let’s introduce all these divisions below.
7 | 17017
———-
11 | 2431
———-
13 | 221
———-
17 | 17
———-
end| 1
Thus, prime factorization of 17017
= 7 x 11 x 13 x 17. Years.
