RO-TST 1987 Problem 2

Find all positive integers $A$ which can be represented in the form: $$A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right)$$ where $m\geq n\geq p \geq 1$ are integer numbers.

Ioan Bogdan