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Greatest common divisor:                                                The greatest common divisor of  two integers a and b.Which are not both zero is the largest integer that divides both a and b.Which is written as (a,b) and is also written as,     gcd(a,b) The  greatest common divisor is a number which should divide the number a and b,Now we find the gcd of some numbers as under. For example:  Find the gcd of  24 and 30. First find the divisors of  24 and 30  we have.         24 = 1,2,3,4,6,8,12,24         30 = 1,2,3,5,6,10,15,30 Now we seen that there are many common divisors of 24 and 30.these are (1,2,3,6) but we have to chose the largest number which is a common divisor of 24 and 30.Here 6 is the largest integer that is the greatest common divisor of both 24 and 30.So 6 is the greatest common divisor of 24 and 30. Another example:      find the gcd of    20 and 35.            Factors of 20 and 35. 20=1,2,4,5,10,20 35=1,5,7,35 So the greatest com

Greatest integer function:                                                   The greatest integer function is a real number  x, denoted by [x] is the largest integer less than or equal to x. The equation of the greatest integer function is:                                         [x] ≦ x < [x] + 1 For example: x= 5.6 Then the greater integer function of x is. [x]=[5.6]=5 Lets do a proof. Show that if n is an integer than [x+n]=n+[x],where x is a real number. solution: proof      Let [x]=m, so m is an integer  this implies that,      m ≦  x < m + 1 now adding n to the inequality we get.     m + n ≦ x +n < m + n +1 This shows that m + n is the largest integer less than or equal to x + n,    [x + n]=m + n               as , m=[x] so,  [x + n]= [x] + n Fractional part:             The fractional part of a real number x,denoted by {x}, is the difference  of  x - [x], Where [x] is the greater integer function. For example: the fri