Question

Find the total number of values of x of the form 1/n,n belongs to N equation
{x}+{2x}+{3x}...{12x}=72x
where {} represents the fractional part function

Answer 1

let us say that x is such that  12 x is also  less than 1.
   x < 1/12   ie.,  x = 1/13 or 1/14  etc..

then  LHS =  {x } + (2x} + {3x} +... + {11 x} + {12 x} = 72 x
  =>  x + 2 x + 3x + ... + 11 x + 12 x  as all these are < 1 and fractional parts.
  =>  LHS = 78 x    it is more than 72 x = RHS.

So x >= 1/12
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let  x = 1/12
    LHS = x + 2x + 3x + .... + 11x + 0,  as fractional part of 12 x is 0.
           =  66 x    This is < RHS.
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let  x = 1/11
   LHS = x + 2x + 3x + 9 x + 10 x + 0 + x        as 11x is integer.  {12 x } = x
           = 56 x  <  RHS 
 
So as x increases from 1/12  towards 1/2,    LHS becomes  smaller.  

There is no n such that the given equation is valid.  - if I understood the question correctly.

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If on the RHS  we have    78 x  then,    the equation is valid for x < 1/12
           then n is  > 12    =>    { x } = {  1/n :  n > 12 }  
 
if on the RHS we have    66 x  then,  equation is valid for x = 1/12
                         then n = 12

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if on RHS  we have    56 x then,    x = 1/11