How to find factors of a number is the question that arises while solving many problems related to HCF, quadratic equations, etc. The solution to this is mastering the methods and types of model questions involved in finding factors of a number.

## Factors of a number Definition

Any number of numbers that divides the given number exactly with leaving zero remainders are factors of a number.

**Example**

What are the factors of a number 15?

Factors of 15 are 1, 3, 5, and 15.

Factors of a Prime Number

For a prime number, its factors are 1 and itself.

**Example for factors of prime number 9, 5, 13, 17 and 19**

Prime Numbers |
Factors |

9 | 1 & 9 |

5 | 1 & 5 |

13 | 1 & 13 |

17 | 1 & 17 |

19 | 1 & 19 |

**Note:**

- The number of factors for any prime number are only 2.
- Since, 1 is having only one factor it is not a prime number.

Factors of Composite Numbers

Composite numbers has more than two factors.

**Examples for factors of composite numbers**

Composite Numbers |
Factors |

10 | 1, 2, 5 & 10 |

15 | 1, 5 & 15 |

30 | 1, 2, 3, 5, 6, 10, 15 & 30 |

Factors of an Algebraic Expression

The values that can exactly divide the algebraic expression are factors of that algebraic expression.

**Examples for factors of an Algebraic Expression**

Algebraic Expression |
Factors |

3abc | 1, 3, a, b, c, 3a, 3b, 3c, ab, ac, bc, 3ab, 3bc, 3ca, 3abc & abc |

**Factor Formula**

There are three formulas for factors

- Number of factors formula
- Sum of factors formula
- Product of factors formula

Let us consider a number ‘N’

Using prime factorisation convert the number to product of prime numbers

I, e N = a^{x} x b^{y} x c^{z}

Here a, b, c are prime numbers

x, y, z are respective powers to those prime numbers.

**How to find number of factors of a number using factor formula?**

The formula below helps in finding the number of factors for number ‘N’

Number of factors formula for N = (x+1)(y+1)(z+1)

**Example Problem **

Calculate the number of factors that can have for the number 60?

**Solution:
**Given number (N) = 60

Factors of 60 = 2

^{2}x 3

^{1}x 5

^{1}

Here prime numbers are 2, 3 & 5

i, e a=2, b=3, c=5

Respective powers are 2, 1 & 1

Using number of factors formula N=(x+1)(y+1)(z+1)

N=(2+1)(1+1)(1+1) => 3 x 2 x 2

=>12

Total number of factors for 60 are 12.

**How to find sum of factors of a number using factor formula?**

Below is the formula for finding sum of factors for number ‘N’

Sum of factors formula for N is

= [( a^{x+1} – 1 )/(a-1)] x [( b^{y+1} – 1 )/(b-1)] x [( c^{z+1} – 1 )/(c-1)]

**Example Problem **

Calculate the sum of factors that can have for the number 60?

**Solution:
**Given number (N) = 60

By prime factorisation, Factors of 60 = 2

^{2}x 3

^{1}x 5

^{1}

Here prime numbers are 2, 3 & 5

i, e a=2, b=3, c=5

Respective powers are 2, 1 & 1

Using number of factors formula N=[( a

^{x+1}– 1 )/(a-1)] x [( b

^{y+1}– 1 )/(b-1)] x [( c

^{z+1}– 1 )/(c-1)]

N=[( 2

^{2+1}– 1 )/(2-1)] x [( 3

^{1+1}– 1 )/(3-1)] x [( 5

^{1+1}– 1 )/(5-1)]

=>[( 2

^{3}– 1 )/(1)] x [( 3

^{2}– 1 )/(2)] x [( 5

^{2}– 1 )/(4)]

=>[( 8- 1 )] x [( 9 – 1 )/2] x [( 25 – 1 )/4]

=>[7] x [8/2] x [24/4]

=>7 X 4 X 6

=>168

The sum of factors for 60 is 168.

**How to find product of factors of a number using factor formula?**

Below formula helps in finding the product of factors for number ‘N’

Product of factors formula for N = N^{(Total number of factors)/2}

= N^{[(a+1)(b+1)(c+1)]/2}

as we know for the number (N) total number of factors = (a+1)(b+1)(c+1)

**Example Problem **

Calculate the product of factors that can have for the number 60?

**Solution:
**Given number (N) = 60

Factors of 60 = 2

^{2}x 3

^{1}x 5

^{1}

Here prime numbers are 2, 3 & 5

i, e a=2, b=3, c=5

we, have already found number of factors for 60 and

it is 12

From product of factors formula =N^{(Total number of factors)/2}

= 60^{12/2}

= 60^{6}

Total number of factors for 60 are 60^{6}.

**Important properties of Factors**

- The two arithematic operations used to find factors of a number are division & multiplication.
- All the number excluding 0 and 1 has atleast two factors i, e 1 and itself.
- For a given number the value of factor will be less than or equal to it.
- For any given number the number of factors are finite.