multiples and factors

Multiples And Factors

Introduction: 
*    If a number ‘a’ divides another number ‘b’ exactly then we say that ‘b’ is called multiple of ‘a’ in other words ‘a’ is called factor of ‘b’.
    Ex :    a = 3, b =45
    clearely \(a \over b\) (a divides b)    
    Then ‘b’ is multiple of ‘a’ (or) ‘a’ is a factor or ‘b’
*    The product of a number and counting numbers are known as the mutliples of the number.
    Ex :    consider the number ‘6’     
    \(6(6\times1),12(6\times2),18(6\times3)......\) are multiples of ‘6’
    \(6=6\times1,6=2\times3\)
    The factors of 6 are 1,2,3,6.

Note 1 : Every number is muliple of itself and also factor of itself.
    2. For every number there are infinite multiples.
    Ex : \(5(5\times1),10(5\times2),15(5\times3)\),...... 

Note : Number of factors of a number are finite.
    Ex : 125 = 125 x 1 
           125 = 5 x 5 x 5
    The number of factors of 125 are 3. (1,5,125).

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
     \(12\over 2\)= 2 x 3 = 6,  \(12 \over 3\)= 2 x 2 = 4 ,\({12 \over 2\times2}={12 \over 4}=3\)
    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
    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.

Perfect number : The number in which sum of all factors is equal to twice the number is called a perfect number.

Illustration : The factors of ‘6’ are 1,2,3,6 1+2+3+6+=12 2 x 6 (twice of 6)
Ex : The factors of 28 are 1,2,4,7, 14,28
       1+2+4+7+14+28= 56 = 2 (28) 

Note : factors of 15 are 1,3,5,15    
    1+3+5+152 x 15 
     15 is not a perfect number