3. EVEN, ODD NUMBERS, PRIME, TWIN PRIME, COPRIME, COMPOSITE NUMBERS

 

EVEN, ODD NUMBERS, PRIME, TWIN PRIME, COPRIME, COMPOSITE NUMBERS

Learn and Remember :

 1. Even Number :

(i) A number (except zero) which is exactly divisible by 2 is called an even number.

(ii) The units place digit of an even number is 0, 2, 4, 6 or 8.

2. Odd Number :

(i) A number, which when divided by 2 gives the remainder 1, is called an odd number.

(ii) The units place digit of an odd number is 1, 3, 5, 7 or 9.

3. (i) The difference between any two consecutive even numbers or odd numbers is 2.

(ii) If any three consecutive even numbers or odd numbers are considered then the

       difference between the greatest number and the smallest number is 4.

4. (i) The sum of any two odd numbers is an even number.

(ii) The product of any two odd numbers is an odd number.

(iii) If an even number is divisible by an odd number then the quotient is an even number.

e.g. 18 (even) ÷  3 (odd) = 6 (even)

(iv) If an odd number is divisible by an odd number, then the quotient is also an odd number.

e.g. 117 (odd) ÷ 13 (odd) = 9 (odd)

5. Prime Numbers :

(i) A number, which is greater than 1 and is divisible only by 1 and the number itself, is called a prime number.

(ii) There are 25 prime numbers between 1 and 100. They are 2, 3, 5, 7, 11, 13,

17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

[Note : The greatest three-digit prime number is 997.]

(iii) 2 is the only even number which is prime. ( imp )

6. Composite Numbers :

(i) Numbers other than l which are not prime numbers are composite numbers. There are 74 composite numbers from 1 to 100.

(ii) 1 is neither a prime number nor a composite number.

7.  Twin Prime Numbers :

(i) If there is one and only one number between two prime numbers, then the two prime numbers are called twin prime numbers.

(ii) There are eight pairs of twin prime numbers between 1 and 100. They are 3, 5; 5, 7;

11, 13; 17, 19; 29, 31; 41, 43; 59, 61; 71, 73.

197, 199; 239, 241; 599, 601 are some more examples of twin prime numbers.

8. Coprime Numbers : Numbers which do not have any common divisor other than 1, are called coprime numbers. Any two consecutive numbers are always coprime numbers. e.g. 25 and 26 or 13 and 35 are coprime numbers.

9. (i) The sum of all the numbers from 1 to 10 is 55. The sum of the even numbers from 1 to 10 is 30, and the sum of the odd numbers from 1 to 10 is 25.

(ii) Thus we can easily find the sum of any ten numbers like 51 to 60, 61 to 70, etc. e.g.

(a) The sum of the numbers from 61 to 70:

61 to 70 = 60 + 1, 60+ 2, ..., 60 + 10.

60 x 10 = 600

Sum of the numbers from 1 to 10 = 55.

∴ the sum of the numbers from 61 to 70 = 600 + 55 = 655.

(b) The sum of the even numbers from 61 to 70 = 330 and the sum of the odd numbers from 61 to 70 = 325.

(c) The sum of the numbers from 931 to 940

= 930 × 10 + 55 = 9355.

[The sum of all natural numbers from 1 to n =n(n+1)/2

∴ the sum of all natural numbers from 1 to 10 =10(10 + 1)/2 = 5× 11=55]