Egyptian Floor Fractions

Algebra Level 5

xx is a real number that satisfies 1x=12x+13x+15x\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 xx be MM, and the smallest possible value of xx be mm. If M+mM+m can be expressed as pq\dfrac{p}{q} for positive coprime integers p,qp,q, then what is p+qp+q?

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