Any number which is an exact divisor of a given number is called a factor of the given number.

For example factor of 6 are 1, 2, 3 and 6

Important results :

(a) 1 is a factor of every number.

(b) Every number is a factor of itself.

(c) Every factor of a number is always equal to or less than the number.

(d) Every number has a finite number of factors.

 

Just as 1, 2, 3 and 6 are factors 6, we say that 6 is multiple of 1, 2, 3 and 6

A number is a multiple of each of its factors

Important results :

(a) Every number is a multiple of itself.

(b) Every multiple of a number is equal to or greater than the number

(c) Every number has as infinite number of multiples.

 

(a) Even Number : A number which is exactly divisible by 2 is called an even number.

 

(b) Odd Number : A number which is not exactly divisible by 2 is called an odd number.

        Example of odd numbers are : 1, 13, 15, 25, 29, ...........

(c) Prime Numbers : A natural number greater than 1, which has no factors except 1 and itself is called a prime number.

        Examples of prime numbers are : 2, 3, 5, 11, 13, 17, ........

 

          Every even number greater than 4 can be expressed as a sum of two odd prime numbers, 

          e.g., 6 = 3 + 3; 18 = 5 + 13; 44 = 13 + 31.

 

(d) Composite Numbers : A number is composite if it has at least one factor other than 1 and itself.

        Example of composite numbers are 4, 6, 8, 9, 10, 12, 14,........

 

1. 1 is neither prime nor composite.

2. Every natural number except 1 is, either a prime number or a composite number.

3. 2 is the only prime number which is even. All other prime numbers are odd.

 

(e) Twin primes : Pairs of prime numbers that have a difference of 2 are called twin primes.

       Example of twin primes are : (3, 5), (5, 7), (11, 13), (17, 19),........

 

(f) Perfect Numbers : If the sum of all the factors of a number is twice the number, then number is called a perfect number. For example, 6 is a perfect number since the factors of 6 are 1, 2, 3, 6 and their sum 1 + 2 + 3 + 6 = 2 × 6.

 

(g) Coprime Number : Two numbers are said to be coprime if they do not have a common factor other than 1.

      Examples of coprime numbers are : (8, 15); (5, 9); (2, 11)

 

1. Two prime numbers are always coprime.

2. Two coprime numbers need not be both prime n
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