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