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Playing with Numbers

Can we exactly divide 26 by 2?

Yes, we can.

Can we exactly divide 26 by 3?

No.

Now, can we say that 2 is a factor of 26 while 3 is not?

Let us find out. Let us first understand the meaning of the term “factor” of a number.

The term “factor” can be defined as follows:

An exact divisor of a number is called a factor of that number.

In the above example, since 2 exactly divides 26 i.e., 2 is an exact divisor of 26, we can say that 2 is a factor of 26. On the other hand, since 3 is not an exact divisor of 26, we can say that 3 is not a factor of 26.

To understand the concept better, look at the following video.

Till now, we were talking about the exact divisors of the numbers (i.e., factors). Let us now look at this from a different point of view.

Let us consider the relation, 24 = 4 × 6

We already know that 4 and 6 are the factors of 24. Apart from this, we can also say that 24 is a multiple of both 4 and 6.

Let us now look at some examples to understand the method of finding factors and multiples better.

Example 1:

Write all the factors of the following numbers.

(a) 8 (b) 18 (c) 64

Solution:

(a) We can write 8 as

1 × 8 = 8

2 × 4 = 8

Therefore, the factors of 8 are 1, 2, 4, and 8.

(b) We can write 18 as

1 × 18 = 18

2 × 9 = 18

3 × 6 = 18

Therefore, the factors of 18 are 1, 2, 3, 6, 9, and 18.

(c) We can write 64 as

1 × 64 = 64

2 × 32 = 64

4 × 16 = 64

8 × 8 = 64

Therefore, the factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

Example 2:

Write the first four multiples of the following numbers.

(a) 6 (b) 9

Solution:

(a) The first four multiples of 6 can be calculated as

6 × 1 = 6

6 × 2 = 12

6 × 3 = 18

6 × 4 = 24

Therefore, the first four multiples of 6 are 6, 12, 18, and 24.

(b) The first four multiples of 9 can be calculated as

9 × 1 = 9

9 × 2 = 18

9 × 3 = 27

9 × 4 = 36

Therefore, the first four multiples of 9 are 9, 18, 27, and 36.

Video(s) by teachers

Look at the following relations.

2 = 2 × 1, 10 = 10 × 1, 37 = 37 × 1, 145 = 145 × 1

Can we make any observation from these relations?

Let us study the properties of numbers with respect to their factors or divisors with the help of these relations. For this, let us go through the following video.


Now, we know the properties of divisors of numbers. Let us discuss some properties of multiples of numbers with the help of the following video.

Let us now look at a few examples to have a better understanding of the concept.

Example 1:

How many multiples of 8 are there in total? Find all the multiples of 8 up to 100.

Solution:

By the property of multiples of a number, we can say that there are infinite multiples of 8.

The multiples of 8 can be calculated as:

8 × 1 = 8

8 × 2 = 16

8 × 3 = 24

8 × 4 = 32

8 × 5 = 40

8 × 6 = 48

8 × 7 = 56

8 × 8 = 64

8 × 9 = 72

8 × 10 = 80

8 × 11 = 88

8 × 12 = 96

8 × 13 = 104

Therefore, the multiples of 8 upto 100 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and 96.

Example 2:

Write the divisor which is common to the numbers 20, 10, 7, and 5.

Solution:

Since 1 is a divisor of all the natural numbers, the common divisor of 20, 10, 7, and 5 is 1.

Example 3:

Write the greatest and the least factor of 42.

Solution:

By the properties of factors of a number, 1 is the least factor and 42 is the greatest factor of 42.

What are the factors of the number 28?

We know that the factor of a number is an exact divisor of that number. Let us now list the factors of the number 28.

They are 1, 2, 4, 7, 14, and 28. Now, let us add all these factors.

1 + 2 + 4 + 7 + 14 + 28 = 56

Can you relate this sum with the original number i.e., 28?

Yes, the obtained sum, i.e. 56, which is twice the original number 28.

Such a number is called a perfect number.

Go through the following video to understand the concept of perfect numbers.

Note: 6 is the smallest perfect number.

Now, are all the numbers perfect numbers?

Let us take an example of 42.

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

Now, sum of its factors = 1 + 2 + 3 + 6 + 7 + 14 + 21 + 42 = 96

However, 96 ≠ 2 × 42

Thus, we can say that 42 is not a perfect number.

From this example, we can conclude that all numbers are not perfect numbers.

Let us now look at a few more examples.

Example 1:

Find the perfect numbers from the following numbers.

(i) 35 (ii) 76 (iii) 496 …

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