DIFFERENCE BETWEEN DIVISORS AND FACTORS

The divisors of a number other than 1 and the number itself are called the factors of that number. 

Let us consider the table given below. 

In the above table, let us consider the number 42.

Divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

Here, other than 1 and 42, the remaining divisors are called factors.  

More clearly, 

2, 3, 6, 7, 14 and 21 are called as factors. 

From the above explanation, we can know the following important fact. 

That is, all the factors are divisors and all the divisors are not factors. 

In the above example, 1 and 42 are divisors, but they are not factors. 

Factors of a Prime Number

Let us consider the prime number 7. 

The divisors of 7 are 1 and 7. 

According to the stuff explained on "Divisors and factors", 7 has no factor. 

So, prime number has no factors and it will have only two divisors. They are 1 and the number itself. 

Factors of a Composite Number

If a number is not prime, then it is composite.

If a number has more than two divisors, then it has factor or factors and it is composite. 

For example, the number 4 has the divisors 1, 2 and 4.

Since it has more than two divisors, the number 4 is composite.

According to the stuff explained on "Divisors and factors", 4 has one factor, that is "2". 

Points to Remember

1)  "1" has only one divisor, that is 1. So 1 is neither prime nor composite.  

2)  1 and number itself are divisors of any number. 

3)  Is there any number which has no divisor ? 

No, because 1 is a divisor of all numbers.

4) Some numbers have more than two divisors. Those numbers are called as composite numbers. 

5) Some numbers have exactly two divisors. Those numbers are called as prime numbers. 

6)  Prime numbers has no factors. 

7) All factors are divisors, but all divisors are not factors. 

Questions on Divisors and Factors

Question 1 :

Which of the following is not a factor of 48?

(i) 6    (ii) 7      (iii) 12      (iv) 48

Solution :

For every number 1 and itself must. Listing out the other factors, by writing it as product of two numbers.

48 = 2 x 24

48 = 3 x 16

48 = 4 x 12

48 = 6 x 8

So, the factors are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

Question 2 :

Which of the following is a prime number?

(i) 91     (ii) 57       (iii) 97     (iv) 81

Solution :

A number which has only two factors 1 and itself is known as prime number.

Factors of 91 :

1, 7, 13 and 91

So, 91 is not a prime number.

Factors of 57 :

1, 3, 19 and 57.

So, 57 is not a prime number.

Factors of 97 :

1 and 97

Since 97 has only two factors 1 and 97. It is a prime number.

Question 3 :

The LCM of 10, 20, 25 is

(i) 20       (ii) 50      (iii) 100     (iv) 75

Solution :

To find the least common multiple of the given numbers, we have to use the ladder method.

LCM = 5 x 2 x 2 x 5

= 100

So, the least common multiple of the given numbers is 100.

Question 4 :

The prime factorisation of 36 is

(i) 4 × 3 × 3 (ii) 2 × 2 × 9 (iii) 12 × 3 (iv) 2 × 2 × 3 × 3

Solution :

By doing the prime factorization of 36

36 = 2 x 2 x 3 x 3

So, option d is correct.

Question 5 :

48 has factors.

(i) 10     (ii) 8     (iii) 7      (iv) 6

Solution :

48 = 6 x 8

So, 6 and 8 are the factors of 48.

Question 6 :

Write the first four multiples of:

(a) 7    (b) 9    (c) 12   (d) 1    (e) 13

Solution :

Multiples of 7 :

7, 14, 21, 28, 35, ........

Multiples of 9 :

9, 18 27, 36, ........

Multiples of 12 :

12, 24, 36, 48,  ........

Multiples of 1 :

1, 1, 1, 1, .........

Multiples of 13 :

13, 26, 39, 52, ........

Fill in the blanks.

Question 7 :

2 × 3 × 7 = 42; 42 is a multiple of ___ ,___ , and ____.

Solution :

Here by multiplying 2, 3 and 7 we get 42. So, 2, 3 and 7 are the factors of 42

Question 8 :

Fifth multiple of 9 : ____

Solution :

Multiples of 9 are 9, 18, 27, 36, 45, ........

So, the 5^th multiple of 9 is 45.

Question 9 :

Is 48 a multiple of 6? (Yes/No)

Solution :

Dividing 48 by 6, we get 8 as quotient and 0 as remainder. Then it must be multiple of 6. So, answer is yes.

Question 10 :

Least multiple of 65 is .

Solution :

65 itself is the least multiple of 65.

Question 11 :

7 × 3 = 21 : ___and ___ are the factors of ___

Solution :

Here by multiplying 7 and 3, we get 21. So 7 and 3 are the factors of 21.

Question 12 :

Is 8 a factor of 70? (Yes/No)

Solution :

Dividing 70 by 8, we get quotient = 8 and remainder = 6

Since the remainder is not equal to 0, we cannot decompose as product of two integers. So, 8 is not a factor of 70.

Question 13 :

(g) Is 1 a prime number?

(Yes/No) 

Solution :

Since 1 is a number which has factor of 1 alone, it is not a prime number. So, No is the answer.

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