Factors and Multiples

Factors

A number which divides the other number exactly is called a factor of that number. In other words, every number is completely divisible by its factors without leaving a remainder. You can check new Factoring Calculator.

Consider 21 ÷ 3

When we divide 21 by 3, the remainder is zero (0). This means, 3 divides 21 exactly. So 3 is a factor of 21.

Here 1,2,3 and 6 are all factors of 6. Similarly, 1 and 19 are factors of 19.

Let us consider  25 ÷ 3

When we divide 25 by 3, the remainder is not zero. This means, 3 does not divide 25 exactly.  So, 3 is not a factor of 25.

None of the numbers 5,7,8,9,10 and 11 is a factor of 12.

Finding a factor of a number:

To find all the factors of a small number is to use factor pairs.
Factor pairs for any number are the pairs of factors that, when multiplied together, yield that integer.

Step by step procedure to find the factors of a number:

For example to find factors of 72:
(1) Make a table with 2 columns labeled “Small” and ‘Large.”
(2) Start with 1 in the small column and 72 in the large column.
(3) Test the next possible factor of 72 (which is 2). 2 is a factor of 72, so write “2” underneath the “1’ in your table. Divide 72 by 2 to find the factor pair: 72 ÷ 2 = 36. Write ‘36” in the large column.
(4) Test the next possible factor of 72 (which is 3). Repeat this process until the numbers in the small and the large columns run into each other. In this case, once we have tested 8 and found that 9 was its paired factor, we can stop.

Small	Large
1 2 3 4 6 8	72 36 24 18 12 9

 

Multiple

A multiple of a number is a number obtained by multiplying it by any integer.
If we multiply 3 by 1,2,3,4,….., we get
3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, …..
Thus, 3,6,9,…. are all multiples of 3.
Clearly, a number is a multiple of each of its factors.
Example: The multiples of 4 are: 4,8,12,16,20,…..
Each of these multiples is greater than or equal to 4.

When we write a number 20 as 20 = 4 × 5, we say 4 and 5 are factors of 20. We also say that 20 is a multiple of 4 and 5.
The representation 24 = 2 × 12 shows that 2 and 12 are factors of 24, whereas 24 is a multiple of 2 and 12.

Integer is always both a factor and a multiple of itself, and that 1 is a factor of every integer.

Properties and facts about Factors and Multiples:

 1) 1 is a factor of every number.
Example: 1 = 1 x 1
2 = 1 x 2
3 = 1 x 3 and so on.
 2) Every number is a factor of itself.
Example: 1 = 1 x 1
6 = 6 x 1
10 = 10 x 1 and so on.
 3) Every factor of a number is an exact divisor of that number.
Example: Factors of 12 are 1,2,3,4,6,12. We find that each one of these factors divides 12. This is true for all other numbers.
 4) Every factor is less than or equal to the given number.
Example: Factors of 34 are 1,2,17 abd 34 itself. Out of these numbers 34 itself is the largest factor and all other factors are less than 34.
The above property is also evident from the fact that divisors of a number are less than or equal to the number itself.
 5) Number of factors of a given number are finite.
Example: By property 4, every factor of a number is less than or equal to the number. Hence, factors of a given number are finite.
 6) Every multiple of a number is greater than or equal to that number.
Example: Multiples of 5 are 5,10,15,…. In fact, multiples of a number are obtained by multiplying the number of 1,2,3,4,5,….. Therefore, the smallest multiple of a number is the number itself. Hence, every multiple of a number is greater than or equal to the number itself.
 7) There are endless multiples of a number.
Example: Multiples of 7 are 7,14,21,28,35,…. Clearly, its a never ending list.
Hence, the number of multiples of a given number is infinite.
 8)Every number is a multiple of itself.
Example: We have, 8 = 8 x 1, 12 = 12 x 1, 15 = 15 x 1, and so on.
So, every number is a multiple of itself.

9) All even numbers are multiple of 2.