Factorization - How to find the factors or prime factors of numbers

FACTORS

A factor or Divisor of any number X is an integer which when multiplied by another factors or divisors produce the original number X. 

Factorization is a process of writing a number as a product of several Divisors or Factors. 

In this post we will learn some quick formulas to calculate factors, prime factors, sum and product of factors etc.

Number of Factors

Let us take any composite number X = Pa x Qb x Rc
Here P, Q, R are prime numbers and a, b, c are their powers respectively.

Formula: (a + 1) x (b + 1) x (c + 1)
For example: 30 can be written as 21 x 31 x 51
No. of Factors will be = (1 + 1)(1 + 1)(1 + 1) = 8

Number of Prime Factors

Convert given numbers into powers of prime numbers. Add the powers of prime numbers.
For example: 12 can be written as 41 x 31 = 22 x 31
No. of Prime factors will be = 2 + 1 = 3

Number of Odd Factors

Convert given numbers into powers of prime numbers. Multiply the power of each odd prime numbers after adding 1 to each power.

If any composite number X = 2a x Qb x Rc, then:
Number of Odd Factors = (b + 1) x (c + 1)

Number of Even Factors

Multiply the number of odd factors by the power of 2.

If any composite number X = 2a x Qb x Rc, then:
Number of Even Factors = a x (b + 1) x (c + 1)

Sum of Factors

If any composite number X = Pa x Qb x Rc, then Sum of factors will be:

Formula: (P0 + P1 + ... + Pa) x (Q0 + Q1 + ... + Qb) x (R0 + R1 + ... + Rc) 

Second Formula: [(Pa+1 - 1)/(P - 1)] x [(Qb+1 - 1)/(Q - 1)] x [(Rc+1 - 1)/(R - 1)]
 
If any composite number X = 2a x Qb x Rc, then:
Sum of Even Factors will be = (21 + ... + 2n) x (Q0 + Q1 + ... + Qb) x (R0 + R1 + ... + Rc)
Sum of Odd Factors will be obtained by removing the powers of 2 = (Q0 + Q1 + ... + Qb) x (R0 + R1 + ... + Rc)

Product of Factors

Formula: (X)(Total divisors)/2

Example: For the given number 40, Find: 
a. Prime factors 
b. Number of Factors or divisors 
c. Sum of factors 
d. Product of factors
e. Number of odd Factors
f.  Number of even Factors
Solution: 40 = 2 x 2 x 2 x 5
                        = 23 x 51                             
a.) Prime Factors = 3 + 1 = 4 (2, 2, 2, 5)

b.) Number of Factors or Divisors = (3 + 1) x (1 + 1) = 8
Eight Factors are: (1, 2, 4, 5, 8, 10, 20, 40)

c.) Sum of Factors = (20 + 21 + 22 + 3) x (50 + 51) = 90
To find sum we can also add Factors = 1 + 2 + 4 + 5 + 8 + 10 + 20 + 40 = 90

d.) Product of Factors = (40)(8)/2 =  (40)4

e.) Number of Odd Factors = 1 + 1 = 2

f.) Number of Even Factors = 3 x 2 = 6

These are just quick formulas to find number of factors, number of prime factors, sum and product of factors.Good Luck 👍