Multiples And Factors

Factor trees :

The diagrams given here, are called as “factor trees”.
The factors of a number. A factor divides its mutliple exactly (without remainder) for example let us consider the numbers 6,12 and 24
            

Look at the different factor trees of 30  
               
 
The bottom row of each of the trees shows that the factors present cannot be split up any further.
Such numbers which cannot be factorized are called as prime numbers

 
Note :
i) The numbers that are multiplied to get the product are called the factors of the product
          Ex : 12 = 2 x 2 x 3 ,    12 = 4 x 3
                   12 = 12 x 1         12 = 6 x 2
           Factors of 12 are 1,2,3,4,6,12

ii) When two or more numbers are multipled, then each number is a factor of that product 
    Ex : 24 = 2 x 2 x 2 x 3
    2 x 2 = 4,     2 x 2 x 2 = 8, 2 x 3 = 6 etc are factors of 24.

iii) Every factor of a number is an exact divisor of that number.
    Ex : 12 = 2 x 2 x 3
    \(\frac {12} {2}\) = 2 x 3 = 6, \(\frac {12} {3}\) = 2 x 2 = 4 , \( \frac{{12}} {{2 \times 2}} = \frac{{12}} {4} = 3\)

iv) Every factor of a number is less than or equal to that number
    6 = 2 x 3         6 = 1 x 6
    2 < 6, 3 <6        1 < 6

v) Number of factors of a given number is finite.
     24 = 24 x 1 ,         24 = 12 x 2 , 24 = 8 x 3 
    24 = 2 x2 x 2 x 3 ,         24 = 4 x 6
\(\therefore\) The number of factors are (1,2,3,,4,6,8,12,24)

vi) ‘1’ is the least factor of any number

vii) Greatest factor for any number is the number itself.
  \( x \in N \Rightarrow x\,\,\)  is a factor of itself.