The statement $3 \times 4 = 12$ has two parts: the numbers that are being multiplied, and the answer. The number that are being multiplied are factors, and the answer is the product. We say that $3$ and $4$ are factors of $12$.

The factors of a number are those number whose continued product is the number: thus $4$ and $2$ are factors of $8$; $3$ and $6$ or $3$, $3$ and $2$ are the factors of $18$. Factors can also be used to determine how quantities can be divided into smaller groups. In this section we will be learning more about factors of a number.

If $a$ and $b$ represent integers, then $a$ is said to be a factor of $b$ if $a$ divides $b$ evenly, that is, if $a$ divides $b$ with no remainder.

The prime factors of a number are those prime numbers whose continued product is the number; thus the prime factors of $12$ are $2$, $2$ and $3$.
The factors of a number can be found by two methods:
  1. Using division
  2. Using multiplication
1. Factorization using division:

In this method, start dividing the given number by $1$, $2$, $3$ and so on to determine which number can divide the given number exactly without giving any remainder. When the remainder is zero, both the divisor and the quotient are the factors of the number. When u reach a point where the factors begin to be repeated, there will not be any further factors.
For example find the factors of $20$

$20 \div 1 = 20$ … $1$ and $20$ are factors of $20$

$20 \div 2 = 10$ … $2$ and $10$ are factors of $20$

$20 \div 3 =$ Leaves reminder

$20 \div 4 = 5$ … $4$ and $5$ are factors of $20$

$20 \div 6 =$ Leaves reminder

$20 \div 7 = $ Leaves reminder

$20 \div 8 = $ Leaves reminder

$20 \div 9 = $ Leaves reminder

$20 \div 10 = 2$ … $10$ and $2$ are factors of $20$

Stop here, since $10$ and $2$ are already the factors.

$\therefore$ the factors of $20$ are $1$, $2$, $4$, $5$, $10$.


2. Factorization by multiplication:

All the numbers used to build a product are the factors of the product.

For example find the factors of $18$.

$1 \times 18 = 18$

$2 \times 9 = 18$

$3 \times 6 = 18$

$\therefore$, the factors of $18$ are $1, 2, 3, 6, 9$ and $18$.
The following are examples of factors of a number.

Solved Examples

Question 1: Is 8 is a factor of 90?
Solution:
 
In order to find the whether 8 is a factor of 90 or not, divide 90 by 8.

8) 90 (11 
   
    10
      8
      2

Since there is a reminder, 8 is not a factor of 90.

 

Question 2: What are the factors of 36?
Solution:
 
$36 \div 1 = 36$ … $1$ and $36$ are factors of $36$

$36 \div 2 = 18$ … $2$ and $18$ are factors of $36$

$36 \div 3 = 12$ … $3$ and $12$ are factors of $36$

$36 \div 4 = 9$ … $4$ and $9$ are factors of $36$

$36 \div 6 = 6$ … $6$ is a factor of $36$

$36 \div 7 = $  Leaves reminder

$36 \div 8 = $ Leaves reminder

$36 \div 9 = 4$ … $9$ and $4$ are factors of $36$

Stop here, since 9 and 4 are  already the factors.

$\therefore$ the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36 

 

Question 3: Find the factors of 16 by multiplication method.
Solution:
 
we need to multiply two numbers to get the given number.

$1 \times 16 = 16$  
$2 \times 8 = 16$
$4 \times 4 = 16$
 
$\therefore$ the factors of 16 are 1, 2, 4, 8, 16.