Playing With Numbers

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:

1. All prime numbers are odd.
2. 2 is the only even prime number.
3. All even numbers are composite numbers.
4. The product of two even numbers is always even.
5. If an even number is divided by 2, the quotient is always odd.
6. Prime numbers do not have any factors.
7. Sum of two prime numbers is always even.
8. The product of three odd numbers is odd.
9. The sum of three odd numbers is even.
10. The sum of two odd numbers and one even number is even.

Exercise 2. Fill in the blanks:

1. The smallest prime number is …………………………
2. The smallest composite number is …………………………
3. The smallest even number is …………………………
4. 1 is neither ………………………… nor…………………………
5. A number which has only two factors is called a …………………………
6. 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.