**Prime and Composite Numbers**

**Prime Numbers**. *An integer* (> 1) *whose only positive factors are* 1 *and that number, is a prime number*.

**Composite Numbers**. *Numbers that have more than two factors are called composite numbers.*

**Co-prime Numbers**. *Two numbers with only* 1 *as a common factor are called co-prime numbers.*

**Twin primes**. *Two prime numbers whose difference is *2 *are called twin primes.*

**Following facts are important:**

- Number 1 is neither prime nor composite.
- The number 2 is the smallest prime number and is even.
- Every prime number other than 2 is odd.
- A number divisible by two co-prime numbers is divisible by their product also.

The method of **Eratosthenes**

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

List of first 100 prime numbers are given below. They are encircled.

**Even Numbers**. *The numbers exactly divisible by *2* are known as even numbers.*

**Odd Numbers**. *The numbers which are not exactly divisible by *2* are known as odd numbers.*

**Notable facts**

- 2 is the smallest prime number which is even.
- Every prime number except 2 is odd.

**Exercise 1**. State whether the following statements are True or False:

- All prime numbers are odd.
- 2 is the only even prime number.
- All even numbers are composite numbers.
- The product of two even numbers is always even.
- If an even number is divided by 2, the quotient is always odd.
- Prime numbers do not have any factors.
- Sum of two prime numbers is always even.
- The product of three odd numbers is odd.
- The sum of three odd numbers is even.
- The sum of two odd numbers and one even number is even.

**Exercise 2**. Fill in the blanks:

- The smallest prime number is …………………………
- The smallest composite number is …………………………
- The smallest even number is …………………………
- 1 is neither ………………………… nor…………………………
- A number which has only two factors is called a …………………………
- A number which has more than two factors is called a …………………………

**Exercise 3**. What is the greatest prime number between 1 and 20?

**Exercise 4**. Write down separately the prime and composite numbers less than 50.

**Exercise 5**. What is the sum of any two (a) Odd numbers? (b) Even numbers?

**Exercise 6**. Justify which among the numbers (a) 23 (b) 51 (c) 37 (d) 26 are prime?

**Exercise 7**. Express (a) 18 (b) 24 (c) 36 (d) 44 as the sum of two odd primes.

**Exercise 8**. Express (a) 21 (b) 31 (c) 53 (d) 61 as the sum of three odd primes.

**Exercise 9**. Give ten pairs of prime numbers whose difference is 2.

**Exercise 10**. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.

**Exercise 11**. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 110.

**Exercise 12**. Write five pairs of prime numbers less than 20 whose sum is divisible by 5.