NCERT Class VI Mathematics text book for Blind and Low Vision Students.

We are now familiar with the factors of a number. Observe the number of factors of a few numbers arranged in this table.

We find that (a) The number 1 has only one factor (i.e. itself ).

(b) There are numbers, having exactly two factors 1 and the number itself. Such number are 2, 3, 5, 7, 11 etc. These numbers are known as prime numbers.

The numbers other than 1 whose only factors are 1 and the number itself are called Prime numbers.

Try to find some more prime numbers other than these.

(c) There are numbers having more than two factors like 4, 6, 8, 9, 10 and so on. These numbers are known as composite numbers.

Numbers having more than two factors are called Composite numbers.

Is 15 a composite number? Why?
What about 18? 25?

Without actually checking the factors of a number, we can find prime numbers from 1 to 100 with an easier method. This method was given by a Greek Mathematician Eratosthenes, in the third century B.C. Let us see the method. List all numbers from 1 to 100, as shown below.

Step 1 : Cross out 1 because it is not a prime number.
Step 2 : Encircle 2, cross out all the multiples of 2, other than 2 itself, i.e. 4, 6, 8 and so on.
Step 3 : You will find that the next uncrossed number is 3. Encircle 3 and cross out all the multiples of 3, other than 3 itself.
Step 4 : The next uncrossed number is 5. Encircle 5 and cross out all the multiples of 5 other than 5 itself.
Step 5 : Continue this process till all the numbers in the list are either encircled or crossed out.

All the encircled numbers are prime numbers. All the crossed out numbers, other than 1 are composite numbers. This method is called the Sieve of Eratosthenes.

Help yourself
Observe that 2 × 3 + 1 = 7 is a prime number. Here, 1 has been added to a multiple of 2 to get a prime number. Can you find some more numbers of this type?

Example 4

Write all the prime numbers less than 15.

Solution :
By observing the Sieve Method, we can easily write the required prime numbers as 2, 3, 5, 7, 11 and 13.

even and odd numbers

Do you observe any pattern in the numbers 2, 4, 6, 8, 10, 12, 14, ...? You will find that each of them is a multiple of 2.

These are called even numbers. The rest of the numbers 1, 3, 5, 7, 9, 11,... are called odd numbers.

You can verify that a two digit number or a three digit number is even or not.

How will you know whether a number like 756482 is even? By dividing it by 2.

Will it not be tedious?

We say that a number with 0, 2, 4, 6, 8 at the ones place is an even number.

So, 350, 4862, 59246 are even numbers. The numbers 457, 2359, 8231 are all odd. Let us try to find some interesting facts:

(a) Which is the smallest even number? It is 2. Which is the smallest prime number? It is again 2.

Thus, 2 is the smallest prime number which is even.

(b) The other prime numbers are 3, 5, 7, 11, 13, ... . Do you find any even number in this list? Of course not, they are all odd.

Thus, we can say that every prime number except 2 is odd.

EXERCISE 3.2

Q1. What is the sum of any two
(a) Odd numbers?
(b) Even numbers?

A1.
(a) even number
(b) even number

Q2. State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.

A2.
(a) False
(b) True
(c) True
(d) False
(e) False
(f) False
(g) False
(h) True
(i) False
(j) True

Q3. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers upto 100.

A3. 17 and 71, 37 and 73, 79 and 97

Q4. Write down separately the prime and composite numbers less than 20.

A4. Prime numbers : 2, 3, 5, 7, 11, 13, 17, 19
Composite numbers : 4, 6, 8, 9, 10, 12, 14, 15, 16, 18

Q5. What is the greatest prime number between 1 and 10?

A5. 7

Q6. Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18

A6.
(a) 3 + 41
(b) 5 + 31
(c) 5 + 19
(d) 5 + 13
(This could be one of the ways. There can be other ways also.)

Q7. Give three pairs of prime numbers whose difference is 2.
[Remark : Two prime numbers whose difference is 2 are called twin primes].

A7. 3, 5; 5, 7 ; 11, 13

Q8. Which of the following numbers are prime?
(a) 23
(b) 51
(c) 37
(d) 26

A8. (a) and (c)

Q9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.

A9. 90, 91, 92 , 93, 94, 95, 96

Q10. Express each of the following numbers as the sum of three odd primes:
(a) 21
(b) 31
(c) 53
(d) 61

A10. (a) 3 + 5 + 13
(b) 3 + 5 + 23
(c) 13 + 17 + 23
(d) 7 + 13 + 41

(This could be one of the ways. There can be other ways also.)

Q11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5.
(Hint : 3+7 = 10)

A11. 2, 3 ; 2, 13; 3, 17; 7, 13; 11, 19

Q12. Fill in the blanks :
(a) A number which has only two factors is called a ______.
(b) A number which has more than two factors is called a ______.
(c) 1 is neither ______ nor ______.
(d) The smallest prime number is ______.
(e) The smallest composite number is _____.
(f) The smallest even number is ______.

A12. (a) prime number
(b) composite number
(c) prime number, composite number
(d) 2
(e) 4
(f) 2