Factors And Multiples

In lower classes, we have learnt multiplication table for numbers. It is time to recollect, to understand the concept of multiples and factors of a number.

FACTORS

When a number is said to be a factor of any other second number, then the first number must divide the second number completely without leaving any remainder. In simple words, if a number (dividend) is exactly divisible by any number (divisor), then the divisor is a factor of that dividend. Every number has a common factor that is one and the number itself.

Factors And Multiples

For example, 4 is a factor of 24 i.e. 4 divides 24 exactly giving 6 as quotient and leaving zero as remainder. Alternatively, 6 is also a factor of 24 as it gives 4 as quotient on division. Therefore, 24 has 1, 24, 4, 6 as its factors in addition to 2, 3, 8 and 12 and all these numbers divide 24 exactly leaving no remainder.

If any natural number has only two factors i.e. 1 and the number itself as its factors, such numbers are called prime numbers. 2 is an example of prime number where it has only two factors 1 and 2.

MULTIPLES

A multiple of a number is a number that is the product of a given number and some other natural number. Multiples can be observed in a multiplication table. Multiples of some numbers are as follows:

Multiples of 2 are

2 × 1 = 2,

2 × 2 = 4,

2 × 3 = 6,

2 × 4 = 8,

2 × 5 = 10,

2 × 6 = 12,

2 × 7 = 14,

2 × 8 = 16,

and so on. Hence, multiples of 2 will be even numbers and will end with 0, 2, 4, 6 or 8.

Multiples of 3 are

3 × 1 = 3,

3 × 2= 6,

3 × 3 = 9,

3 × 4 = 12,

3 × 5= 15,

3 × 6 = 18

and so on.

Multiples of 5 are

5 × 1 = 5,

5 × 2 = 10,

5 × 3 = 15,

5 × 4 = 20,

5 × 5 = 25,

and so on. Every multiple of 5 has its last digit as 0 or 5.

FACTOR vs MULTIPLE

Some may get confused with the concept of multiple and factor. Let us consider an example. In the multiplication table of 6, we have 6 x 3 = 18. Here, 6 and 3 are the factors of 18 while 18 is the multiple of 3 as well as 6. This means both 6 and 3 divides 18 which is same as that 18 is exactly divisible by 6 and 3.

Thus, we can conclude that if X and Y are two numbers and;

•    If X divides Y, X is a factor of Y

•    If Y is divisible by X, Y is a multiple of X

Since the number 1 divides every integer, it is a common factor of every integer. Also, every number is divisible by 1 and every number is a multiple of 1.

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Factors and Multiples:

To understand factors and multiples imagine a situation in which Varsha has 12 marbles and she has to arrange it in different ways. Firstly, she arranges all the marbles in a single row. Next, she arranges the marbles in two rows with 6 marbles in each row. Then she makes 3 rows of 4 marbles each as shown.

Factors and Multiples

Each arrangement consists of 12 marbles and can be represented as product of number of rows and marbles in each row. It can be seen that
12 = 12 × 1

12 = 6 × 2

12 = 4 × 3

It can be seen that on multiplying 1 and 12, 6 and 2, 4 and 3 we get 12. Natural numbers which on multiplication give a new number are known as the factors of the new number. Here, 1, 2, 3, 4, 6 and 12 are factors of 12.

Let’s find out the numbers which divide 6 exactly. By exactly we mean that on dividing 6 by any number, it does not leaves any remainder.

Factors and Multiples

It can be seen that 6 is completely divisible by 1, 2, 3, and 6 itself. These numbers are called factors of 6.

Factors and Multiples

Factors and Multiples

Multiple:

When a number is multiplied by an integer then the product is a multiple of that number.

Example:

Factors and Multiples

7, 14, 21, 28, 35, 42………….. are multiples of 7. Thus, multiples of any number is obtained by multiplying it with integers.

Now try finding yourself, the factors and multiples of 4, 7, 9, 13, 15, 18, 42, 32 and 25.

Factors and Multiples

Common Factors and Multiples:

Let us try to find out the factors of 4 and 12, 15 and 18.

Factors of 4 are 1, 2 and 4 and factors of 12 are 1, 2, 3, 4, 6 and 12.

So, 1, 2 and 4 are common factors of 4 and 12.

Now consider the numbers 4 and 15.

Factors of 4 are 1, 2 and 4 and factors of 15 are 1, 3 and 5.

There is only one common factor and that is 1.

Factors and Multiples

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Practise This Question

Which of the following is the largest two-digit multiple of 19?