MULTIPLES AND FACTORS

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, \(12\over 3\) = 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
    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.
Key Points:
    Every whole number has at least two factors: 1 and itself.
    Factors are always less than or equal to the number itself.
    Zero is a factor of every number, as  0×n=0 for any number n.
    Negative numbers can also be factors, as they can divide certain numbers without leaving a remainder.