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Factors and Multiples

Factors of a number are whole numbers that divide it completely without leaving any remainder.

The whole numbers that multiply to give a third number are the factors of the third number.

Examples: 18 = 1 × 18 18 ÷ 1 = 18

18 = 2 × 9 18 ÷ 2 = 9

18 = 3 × 6 18 ÷ 3 = 6

18 is a composite number. The numbers 1, 2, 3, 6, 9, 18 divide 18 evenly and hence the factors of 18 are 1, 2, 3, 6, 9, 18.

Factoring: The process where a number is written as the product of its factors is called as Factoring. The factors are either prime numbers or composite numbers, except 1 which is neither prime nor composite.

Example: 6 = 3 × 2

6 is factored into 3 and 2

For larger numbers, it is easier to write down the factors in steps by writing down the given large number as a product of two numbers and then again write them down as factors as shown in the example below.

36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6 = 9 × 4

Hence, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Multiples of Numbers

Multiples: The product of any two whole numbers is called as the multiple of the two whole numbers.

Example: 8 × 6 = 48

48 is a multiple of 8 as 8 divides 48 evenly and it is also a multiple of 6 as 6 also divides 48 evenly.

48 is a multiple of both 8 and 6

8 and 6 both are factors of 48

Factors and Multiples

The multiples of 2 are 2, 4, 6, 8, 10 …..

The multiples of 3 are 3, 6, 9, 12, 15, 18, …….

The multiples of 5 are 5, 10, 15, 20, 25, 30, …..

Examples of Multiples

The diagram below shows how Sam is grouping some numbers.

In one circle there are multiples of 2.

In the other circle there are multiples of 3.

There is a shaded space for numbers that are multiples of both 2 and 3.

The numbers that are not a multiple of 2 or 3 go in the space outside the two circles.

Sam has already placed the numbers 1, 5, 4, 12, and 33 in the correct spaces in the diagram.

Example - Multiples

Write 18, 24, 27, 26, 35 and 14 in their correct places in the diagram shown.

Solution: By the use of multiplication facts we know that

2 is a factor of 26, 14 and hence are placed in the multiples of 2 circle.

3 is a factor of and 27 and hence it is placed in the multiples of 3 circle.

2 and also 3 both are factors of 18 and 24. This is also by the use of multiplication facts. And hence, they are placed in the shaded area.

35 is not a multiple of 2 or 3 and hence it is placed outside the two circles.

Example - solution