A given number can be decomposed into two parts in two different ways, such that the first part of the first decomposition is a given multiple of the first part of the second decomposition, and the second part of the first decomposition is a given multiple of the second part of the second decomposition.
To solve the problem with positive integers, we need to correct it as follows: "and the second part of the second decomposition should be a given multiple of the second part of the first decomposition." In this case, the solution is as follows:

I. decomposition: $m x, \quad a-m x, \quad a-x=n(a-m x), \quad$ from which

II. decomposition: $x, \quad n(a-m x), \quad x=a \frac{n-1}{m n-1}$.