You can't. With hope, for 9 unknown independent variables, you need at least 9 equations, or for some kind of situations, some restrictions. Here, the restriction to be natural numbers seems to be not enough.
Suppose that you know that a + b = 5. Then, you can solve for some "originals" a nor b.
Ok, I see now. What you do was a variation of the partial fraction decomposition. What you need to do, them, it's given a rational number, obtain this decomposition. The normal decomposition uses "división entera" (in spanish, don't know the words in english) operation, where can use the mod operator, but usually it's done with celing.
In the attached, your equivalent procedure, with a pseudo cumulant matrix.