FACTOR

Devi has 6 coins with her. She wants to arrange them in columns in such a way that each
column has the same number of coins. She arranges them in many ways using all the 6 coins.
  Case (i) 1 coin in each column
  number of columns = 6
  Total number of coins = 1 × 6 = 6
   Case (ii) 2 coins in each column
   Number of columns = 3
  Total number of coins = 2 × 3 = 6
   Case (iii) 3 coins in each column  
   Number of columns = 2
   Total number of coins = 3 × 2 = 6
   Case (iv) 6 coins in each column
   Number of column = 1
    Total number of coins = 6 × 1 = 6
These are the only possible arrangements using all the 6 coins.
From these arrangements, Devi observes that 6 can be written as a product of two numbers
in different ways as
6 = 1 × 6 6 = 2 × 3 6 = 3 × 2 6 = 6 × 1
From 6 = 2 × 3 it can be said that 2 and 3 exactly divide 6. So, 2 and 3 are factors of 6.
From the other product 6 = 1 × 6, thus 6 and 1 are also factors of 6.
1, 2, 3 and 6 are the only factors of 6.


DEFINITION OF FACTOR:A number which divides the other number exactly is called a factor of that number.

EXAMPLES:
Number   Factors
12       1, 2, 3, 4, 6, 12
18       1, 2, 3, 6, 9, 18
20       1, 2, 4, 5, 10, 20
24       1, 2, 3, 4, 6, 8, 12, 24

PRIME AND COMPOSITE NUMBERS:

These numbers whose only factors are 1 and the number itself
are called prime numbers.

Numbers having more than two factors are called composite numbers.

The number 1 has only one factor (i.e. itself) so, 1 is neither prime nor composite.

CO-PRIMES:

the numbers which have only 1 as the common factor are called co-primes or
relatively prime. 

Example-1. Consider two co-prime numbers 4 and 5. Are both of them prime numbers?
Solution: No, 4 is not a prime. Only 5 is a prime.
We can say that "Only two primes are co-primes but all the co-primes need
not be primes."

TWIN PRIMES:
Twin primes are prime numbers that differ from each other by two e.g. (3, 5), (5, 7),
(11, 13), (41, 43) etc.


PRIME FACTORIZATION

When a number is expressed as a product of its factors, we say that the number has been
factorized. The process of finding the factors is called factorization.
There may be several ways in which a number can be factorized. 
For example, the number
24 can be factorized as:
i) 24 = 1 × 24 ii) 24 = 2 × 12 iii) 24 = 3 × 8
iv) 24 = 4 × 6 v) 24 = 2 × 2 × 2 × 3
In (ii) and (iii) one factor is prime, and the other factor is a composite number. In (iv) both
the factors are composite numbers. However in (v) all the factors are prime numbers. In (i) one
factor is composite.
Factorization of the type (v), where all the factors are prime numbers, is known as prime
factorization.

METHODS OF PRIME FACTORIZATION:

1. Division Method : Prime factorisation of 42 using division method we
proceed as follow:
Start dividing by the least prime factor. Continue division till the resulting
number to be divided is 1.
  Prime factorisation of 42 is 2 × 3 × 7

2. Factor Tree Method : We can find the prime factorization of 60 by drawing a factor tree.
To find the prime factorization of 60 using factor tree method, we proceed as follow:
Step-1:Express 60 as a product of two numbers.
Step-2:Factorise 4 and 15 further, since they are
composite numbers.
Step-3:Continue till all the factors are prime numbers.


COMMON FACTORS

Number                      12   18
Factors of the number       1, 2, 3, 4, 6, 12 1, 2, 3, 6, 9, 18
Common factors of 12 and 18 are 1, 2, 3 and 6