The inoffensive floor function

Algebra Level 5

For any real number \( x\), let \( \left\lfloor x \right\rfloor \) represent the largest integer number less than or equal to \( x\). If \( 1< x<{ 2 }^{ 2015 }\) and \( 2^{2015}\) is a factor of \(\left\lfloor { 2 }^{ 2015 }x \right\rfloor -\left\lfloor x \right\rfloor \), find the exact value of the sum of two coprime numbers \(a\) and \(b\) if \[ \frac { a }{ b } =\sqrt [ 2015 ]{ \frac { \left\lfloor { 2 }^{ 2015 }x \right\rfloor -\left\lfloor x \right\rfloor }{ \left\lfloor x \right\rfloor } } . \]

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