Shortcut to Find Number of Factors of a Number | Sum of Factors of a Number

Table of Contents

How to Find Total Number of Factors for Big Numbers Easily | Number System

In the number system, the concept of factors of numbers is one of the important sub-topics. In this article, we will discuss about definition of factors of number, formulas for finding a number of factors, sum of factors, product of factors, even number of factors, odd number of factors, perfect square factors, and perfect cube factors for any number.

Definition of Factors of a number:

Factors of a number any number “ P” refers to all the numbers which are exactly divisible on “P” i.e remainder comes to zero. These factors of numbers are also called divisors of a number.

For example:

Factors of the number 9

1 × 9 = 9,
Also 3 × 3 = 9

So 1, 3, and 9 are factors of 9.

And also -1, -3, and -9 because you get a positive number when you multiply two negatives,

such as (-3)×(-3) = 9

No answer is 1, 3, 9, -1, -3, & -9

But usually, factors of numbers consider only positive numbers

Note: fractions of numbers also not considered as factors

Formulas related to factors of numbers:

Total number of factors
Sum of factors
Product of factors

Take any number “N” and it is to be covert into a product of prime numbers (Prime factorization) i.e

N = A^p x B^q x C^r here A, B , C are prime numbers and p,q, and r were respective powers of those prime numbers.

Total numbers of factors for ” N “= (p + 1)(q +1)(r +1)

The sum of all factors of “N” $= \frac{A^{p+1} - 1}{A - 1} \times \frac{B^{q+1} - 1}{B - 1} \times \frac{C^{r+1} - 1}{C - 1}$

Product of all factors of “N” = ( N )^{Total no. of factors/2}

Example – 1 : Find the number of factors of 98 and also find the sum and product of all factors

Solution : First write the number 98 into prime factorization

98 = 2 x 49 = 2x 7 x 7

98 = 2¹ x 7²Here A = 2 , B = 7 , p= 1 , q = 2

Number of factors for the number 98 = (p + 1)(q +1) = 2 x 3 = 6

Sum of all factors of 98 = $= \frac{2^{1+1} - 1}{2 - 1} \times \frac{7^{2+1} - 1}{7 - 1}$
= 3 x 57 = 171

Product of all factors of number 98 = (98)^6/2 = (98)³ = 941192

Example – 2 : Find the number of factors of 588 and also find the sum and product of all factors

Solution : First write the number 588 into prime factorization

588 = 2 x 294 = 2x 2 x 147 = 2 x 2 x 7 x 21 = 2 x 2 x 7 x 7 x 3

588 = 2² x 3¹ x 7²Here A = 2 , B = 3 , C = 7 , p= 2 , q = 1 and r =2

Number of factors = (p + 1)(q +1)(r +1) = 3 x 2 x 3 = 18

Sum of all factors of 588 $= \frac{2^{2+1} - 1}{2 - 1} \times \frac{3^{1+1} - 1}{3 - 1} \times \frac{7^{2+1} - 1}{7 - 1}$

= 7 x 4 x 57 = 1596

Product of all factors of number 588 = (588)^18/2 = (588)⁹

Another concept in Factors of numbers

How many factors are odd
How many factors are even
Number of perfect square factors
Number of perfect cube factors

Example – 3 : Find the number of odd, even, perfect square, and perfect cube factors of 4500

Solution: First write the number 4500 into prime factorization

4500 = 45 x 100 = 9 x 5 x 10 x 10 = 3 x 3 x 5 x 5 x 2 x 5 x 2

4500 = 2² x 3² x 5³ Here consider A = 2 , B = 3 , C = 5 , p= 2 , q = 2 and r = 3

Here identifying that odd numbers are 3 and 5

Numbers of odd factors of number 4500 = (q + 1 ) (r + 1) = 3 x 4 = 12

Total number of factors = (p + 1)(q +1)(r +1) = 3 x 3 x 4 =36

Numbers of even factors of number = Total number of factors – Numbers of odd factors = 36 – 12 = 24

Number of perfect square factors of number 4500 = 2 x 2 x 2 = 8

( 2² $\overset{F}{\rightarrow}$ 2⁰ , 2²; 3² $\overset{F}{\rightarrow}$ 3⁰ , 3²& 5² $\overset{F}{\rightarrow}$ 5⁰ , 5²)

Number of perfect cube factors of number 4500 = 1 x 1 x 2 = 2

( 2² $\overset{F}{\rightarrow}$ 2⁰3² $\overset{F}{\rightarrow}$ 3⁰ , & 5² $\overset{F}{\rightarrow}$ 5⁰ , 5³)

Example – 4 : Find the number of odd, even, perfect square, and perfect cube factors of 5040

Solution: First write the number 5040 into prime factorization

5040 = 504 x 10 = 4 x 126 x 5 x 2 = 2 x 2 x 18 x 7 x 5 x 2 = 2 x 2 x 3 x 3 x 2 x 7 x 5 x 2

5040 = 2⁴ x 3² x 7¹ Here consider A = 2 , B = 3 , C = 7 , p= 4 , q = 2 and r = 1

Here identifying that odd number are 3 and 7

Numbers of odd factors of number 5040 = (q + 1 ) (r + 1) = 3 x 2 = 6

Total number of factors = (p + 1)(q +1)(r +1) = 5 x 3 x 2 = 30

Numbers of even factors of number = Total number of factors – Numbers of odd factors = 30 – 6 = 24

Number of perfect square factors of number 5040 = 3 x 2 x 1 = 6

( 2² $\overset{F}{\rightarrow}$ 2⁰ , 2², 2⁴; 3² $\overset{F}{\rightarrow}$ 3⁰ , 3²& 7¹ $\overset{F}{\rightarrow}$ 7⁰ )

Number of perfect cube factors of number 5040 = 2 x 1 x 1 = 2

( 2² $\overset{F}{\rightarrow}$ 2⁰ , 2³; 3² $\overset{F}{\rightarrow}$ 3⁰ , & 7² $\overset{F}{\rightarrow}$ 7⁰ )

Related Topics :

Number Categories

Rules for Divisibility of Numbers

Methods to find HCF & LCM

GCD and LCM Problems & Solutions

Formulas for Sum of n Consecutive numbers

Hi friends Thanks for reading. I hope you liked this article ” Shortcut to Find Number of Factors of a Number”. Give feedback, comments and please don’t forget to share it.