NCERT Class 6 mathematics text book for blind and visually impaired students.

Observe the factors of some numbers taken in pairs.

(a) What are the factors of 4 and 18?
The factors of 4 are 1, 2 and 4.
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The numbers 1 and 2 are the factors that are common to both 4 and 18.
They are the common factors of 4 and 18.

(b) What are the common factors of 4 and 15?
These two numbers have only 1 as the common factor.
What about 7 and 16?

Help yourself

Find the common factors of
(a) 8, 20
(b) 9, 15

Two numbers having only 1 as a common factor are called co-prime numbers. Thus, 4 and 15 are co-prime numbers.

Are 7 and 15, 12 and 49, 18 and 23 co-prime numbers?

(c) Can we find the common factors of 4, 12 and 16?
Factors of 4 are 1, 2 and 4.
Factors of 12 are 1, 2, 3, 4, 6 and 12.
Factors of 16 are 1, 2, 4, 8 and 16.
Clearly, 1, 2 and 4 are the common factors of 4, 12, and 16.

Find the common factors of
(a) 8, 12, 20
(b) 9, 15, 21.

Let us now look at the multiples of more than one number taken at a time.

(a) What are the multiples of 4 and 6?
The multiples of 4 are 4, 8, 12, 16, 20, 24, ... (write a few more)
The multiples of 6 are 6, 12, 18, 24, 30, 36, ... (write a few more)
Out of these, are there any numbers which occur in both the lists?
We observe that 12, 24, 36, ... are multiples of both 4 and 6.
Can you write a few more?
They are called the common multiples of 4 and 6.

(b) Find the common multiples of 3, 5 and 6.
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, ...
Multiples of 6 are 6, 12, 18, 24, 30, ...
Common multiples of 3, 5 and 6 are 30, 60, ...

Write a few more common multiples of 3, 5 and 6.

Example 5

Find the common factors of 75, 60 and 210.

Solution :
Factors of 75 are 1, 3, 5, 15, 25 and 75.
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60.
Factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.
Thus, common factors of 75, 60 and 210 are 1, 3, 5 and 15.

Example 6

Find the common multiples of 3, 4 and 9.

Solution :
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, ....
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48,...
Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, ...
Clearly, common multiples of 3, 4 and 9 are 36, 72, 108,...

EXERCISE 3.4

Q1. Find the common factors of :
(a) 20 and 28
(b) 15 and 25
(c) 35 and 50
(d) 56 and 120

A1.
(a) 1, 2, 4
(b) 1, 5
(c) 1, 5
(d) 1, 2, 4, 8

Q2. Find the common factors of :
(a) 4, 8 and 12
(b) 5, 15 and 25

A2.
(a) 1, 2, 4
(b) 1, 5

Q3. Find first three common multiples of :
(a) 6 and 8
(b) 12 and 18

A3.
(a) 24, 48, 72
(b) 36, 72, 108

Q4. Write all the numbers less than 100 which are common multiples of 3 and 4.

A4. 12, 24, 36, 48, 60, 72, 84, 96

Q5. Which of the following numbers are co-prime? (a) 18 and 35
(b) 15 and 37
(c) 30 and 415
(d) 17 and 68
(e) 216 and 215
(f) 81 and 16

A5. (a), (b), (e), (f )

Q6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?

A6. 60

Q7. A number is divisible by 12. By what other numbers will that number be divisible?

A7. 1, 2, 3, 4, 6