NCERT Solutions for Class 6 Maths Chapter 3 Exercise 3.1

Class VI Mathematics NCERT Book Ex. 3.1 of chapter 3 Playing with Numbers updated for CBSE Board and State board student academic session 2024-25. Videos related to 6th Math NCERT ex. 3.1 is also given in solution as well as explanation format, where each question is explained properly.

Factor:
A factor of a number is an exact divisor of that number. In other words, a factor of a number is that number which completely divides the number without leaving a remainder.
Example: We know that, 12 = 4 × 3, 12 = 2 × 6, also, 12 = 1 × 12
This shows that each of the numbers 1, 2, 3, 4, 6 and 12 exactly divides 12.
Therefore, 1, 2, 3, 4, 6 and 12 are all factors of 12.

A multiple of a number is a number obtained by multiplying it by a natural number.
If we multiply 2 by 1, 2, 3, 4, … , we get 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8, … .
Thus, 2, 4, 6, 8, … are multiples of 2.
Clearly, a number is a multiple of each of its factors.

Property 1: 1 is a factor of every number
We have, 1 = 1 × 1; 2 = 1 × 2; 3 = 1 × 3; 9 = 1 × 9 and so on. Thus, 1 is a factor of every number.

We have, 1 = 1 × 1; 4 = 4 × 1; 12 = 12 × 1; 67 = 67 × 1 and so on.
We can express every number in this way. Hence, every number is a factor of itself.

Factors of 21 are 1, 3, 7 and 21 itself. Out of these numbers 21, itself is the largest factor and all other factors are less than 21.
The above property is also evident from the fact that divisors of a number are less than or equal to the number itself.

By property 3, every factor of a number is less than or equal to the number.
Hence, factors of a given number are finite.

Multiples of 3 are 3, 6, 9, 12, 15, ….
In fact, multiplies of a number are obtained by multiplying the number by 1, 2, 3, 4, 5, …. Therefore, the smallest multiple of a number is the number itself. Hence, every multiple of a number is greater than or equal to the number itself.

We have, 5 = 5 × 1; 13 = 13 × 1; 18 = 18 × 1 and so on.
So, every number is a multiple of itself.

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, …. Clearly, it is a never ending list.
Hence, the number of multiples of a given number is infinite.

Even Numbers:
All multiples of 2 are called even numbers.
Example: 2, 4, 6, 8, 10, 12, 14 etc. are all even numbers.

Numbers which are not multiples of 2 are called odd numbers.
Example: 1, 3, 5, 7, 9, 11, 13, 15 etc. are all odd numbers.

If the sum of all the factors of a number is two times the number, then the number is called a perfect number.
Example: 6 is a perfect number, since the factors of 6 are 1, 2, 3, 6 and (1 + 2 + 3 + 6) = (2 × 6). 28 is a perfect number, since the factors of 28 are 1, 2, 4, 7, 14, 28 and (1 + 2 + 4 + 7 + 14 + 28) = (2 × 28).

Each of the numbers which has exactly two factors, namely, 1 and itself, is called a prime number.
Example: The numbers 2, 3, 5, 7, 11, 13, 17 etc. are all prime numbers.

Two numbers are said to be co-prime numbers if they do not have a common factor other than 1.
Example: (2, 3); (3, 4); (5, 6); (8, 13) etc. are pairs of co-prime numbers.

Note: Any two prime numbers are always co-primes, but two co-primes need not be both prime numbers. For example, 14, 15 are co-primes, while none of 14 and 15 is a prime number.

Numbers having more than two factors are known as composite numbers.
Each of the numbers 4, 6, 8, 9, 10, 12, 14, 15, …are composite numbers.

Two prime numbers which have only one composite number between them are called twin primes.
Example: 3, 5; 5, 7; 11, 13 and 17, 19 etc. are pairs of twin primes between 1 and 20.

A set of three consecutive prime numbers which have only one composite number between each two of them are called prime triplet.
Example: 3, 5, 7 is the only prime triplet.

• (i) 1 is neither prime nor composite.
• (ii) 2 is the smallest prime number.
• (iii) 2 is the only even prime number.
• (iv) All other even numbers are composite numbers.