Egyptian Floor Fractions

Algebra Level 5

\(x\) is a real number that satisfies \[\dfrac{1}{\lfloor x \rfloor}=\dfrac{1}{\lfloor 2x \rfloor}+\dfrac{1}{\lfloor 3x \rfloor}+\dfrac{1}{\lfloor 5x \rfloor}\]

Let the largest possible value of \(x\) be \(M\), and the smallest possible value of \(x\) be \(m\). If \(M+m\) can be expressed as \(\dfrac{p}{q}\) for positive coprime integers \(p,q\), then what is \(p+q\)?

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