# Factors of a Number

**Factors of a number** are defined as numbers that divide the original number evenly or exactly. The meaning of a factor is a whole number that can divide a greater number evenly. A factor cannot be a fraction. Each prime number will have only two factors, i.e. 1 and the number itself, whereas all composite numbers will have more than two factors, that include prime factors also. Examples are:

- Factors of 2 = 1 and 2
- Factors of 10 = 1,2,5 and 10

Factors of an algebraic expression are the values that can divide the given expression. For example, factors of 3xyz are 3, x, y, z, 3x, 3y, 3z, xyz and 3xyz. Learn factorisation of algebraic expressions in detail here.

## What are Factors of a Number?

By definition of factors, we know, they are the values that divide the original number into equal parts or numbers. For example, if 9 is the factor of 81, then if we divide 81 by 9, we get:

81 ÷ 9 = 9

Hence, 9 divides 81 into 9 equal parts.

Or

9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 81

**Point to remember:** Fractions cannot be considered as factors for any number.

## Factors of Prime Numbers

The factors of a prime number are only two, 1 and the number itself. We cannot divide prime number by any other number, that can give a whole number. 1 is not a prime number, since it has only one factor.

These prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.

## Factors of Composite Numbers

The composite numbers are those numbers that have factors more than 2. The prime factors of composite numbers are those factors that are prime numbers such as 2, 3, 5, 7, 11, etc.

Composite Numbers |
Factors |

4 | 1,2 and 4 |

8 | 1, 2, 4 and 8 |

14 | 1, 2, 7 and 14 |

20 | 1, 2, 4, 5, 10 and 20 |

35 | 1,5,7 and 35 |

90 | 1,2,3,5,6,9,10,15,18,30,45 and 90 |

## Factors of Square Numbers

Square numbers are those that produced when a number is multiplied by itself. It is represented as n x n = n^{2}, where n is any integer.

2 x 2 = 2^{2} = 4

3 x 3 = 3^{2} = 9

5 x 5 = 5^{2} = 25

10 x 10 = 10^{2} = 100

The above examples prove that one of the factors of a square number is the value, that is square to produce the original number.

## Factors Formulas

There are basically three types of formulas considered for factors. They are:

- Number of Factors
- Product of Factors
- Sum of Factors

Let us assume N is a natural number, for which we need to find the factors. If we convert N into the product of prime numbers by prime factorization method, we can represent it as;

N = X^{a} × Y^{b} × Z^{c}

where X, Y and Z are the prime numbers and a, b and c are their respective powers.

### Number of Factors

The formula for the total number of factors for a given number is given by;

**Total Number of Factors for N = (a+1) (b+1) (c+1)**

### Sum of Factors

The formula for the sum of all factors is given by;

**Sum of factors of N = [(X**^{a+1}-1)/X-1] × [(Y^{b+1}-1)/Y-1] × [(Z^{c+1}-1)/Z-1]

### Product of Factors

The formula for the product of all factors is given by;

**Product of factors of N = N**^{Total No. of Factors/2}

**Example: Find the total number of factors of 90 along with sum and product of all factors.**

Solution: Write the prime factorization of 90 first.

90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5

90 = 2^{1} × 3^{2} × 5^{1}

Here, X = 2, Y = 3, Z =5 and a = 1, b = 2, c = 1

Therefore, total number of factors of 90 = (a +1)(b+1)(c+1) = (1+1)(2+1)(1+1) = 2 × 3 × 2 = 12

Sum of factors of 90 = [(2^{1+1}-1)/2-1] × [(3^{2+1}-1)/3-1] × [(5^{1+1}-1)/5-1] = (3/1) × (26/2) × (24/4) = 3 × 13 × 6 = 234

Product of factors of 90 = 90^{Total factors of 90/2 }= 90^{12/2 }= 90^{6}

## How to Find Factors of a Number?

Knowing how to calculate factors of a number is extremely crucial in Maths. The steps to find the factors of a number are given below in a very easy to understand way. An example is taken to make the explanation easier.

Find the factors of 16.

Step 1: Write the numbers from 1 to 16

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16

Step 2: Check now, which number can divide the 16 equally from the above list of numbers.

16/1 = 16

16/2 = 8

16/3 = fraction

In the same way, divide 16 by each of these numbers.

Step 3: Hence, we get the factors of 16 are 1,2,4,8, and 16

### Shortcut Method to Find Factors of a number

Let us see an example, to find the factors of a number using the shortcut method.

Consider the number as 40.

40 = 10 × 4

= (5 × 2) × 4

=(5 × 2) × (2 × 2)

= (5 × 2) × (2 × 2)

Now, the factors of 40 will include all the combinations from 5 × 2 × 2 × 2 and 1 itself (as 1 × 40 = 40). So, the positive factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. It should be noted that there will also be negative factors whose count has to be even.

## How to Find Factors of Large Numbers?

To calculate the factors of large numbers, divide the numbers with the least prime number, i.e. 2. If the number is not divisible by 2, move to the next prime numbers, i.e. 3 and so on until 1 is reached. Below is an example to find the factors of a large number.

**Example: **1420

Steps |
Prime Factors |
Product |

Step 1: Divide by 2 | 2 | 710 |

Step 2: Again Divide by 2 | 2 | 355 |

Step 3: Divide by | 5 | 71 |

In step 3, a prime number is obtained as a product, and so, the process is stopped. The factors will be all the multiples of 1, 2, 2, 5, 71, 355, 710. Now, the positive factors of 1420 will be 1, 2, 4, 5, 10, 20, 71, 142, 284, 355, 710, and 1420.

In the same case, if only prime factors are considered, it is called the prime factorization of that number. In this way, it is easy to factor a number and know its factors and prime factors.

## Prime Factors of a Number

With the help of prime factorisation method, we can determine the prime factors of a number.

Example: Prime factorisation of 144.

Steps | Prime factors | Division |

Divide 144 by 2 | 2 | 144 ÷ 2=72 |

Divide 72 by 2 | 2 | 72 ÷ 2 = 36 |

Divide 36 by 2 | 2 | 36 ÷ 2 = 18 |

Divide 18 by 2 | 2 | 18 ÷ 2 = 9 |

Divide 9 by 3 | 3 | 9 ÷ 3 = 3 |

Divide 3 by 3 | 3 | 3 ÷ 3 = 1 |

Thus, 144 = 2 × 2 × 2 × 2 × 3 × 3

## List of Factors of Numbers

Below is a list of common numbers with their factors and prime factors. Each of these links will include the process of factoring of any number along with all the factors, including the prime factors of that number.

## Solved Examples

**Q.1: What are the factors of 8?**

Solution: By the definition of factors of a number, we know, 1 and the number itself are the two general factors.

8 is an even number and divisible by 2.

8/2 = 4

also, 8/4 = 2

Apart from these numbers, 8 is not evenly divisible from any other number.

So, 1, 2, 4 and 8 are the required factors.

**Q.2: How many factors do 31 has?**

Solution: Since 31 is a prime number, therefore the factor of 31 are 1 and 31.

Therefore, there are only two factors.

**Q.3: Fill in the blanks: Factors of 9 are 1, ___, 9.**

Solution: Factors of 9 are 1, 3, 9.

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