Linear equation so that the solutions has to be integers.

Question

I have this equation

$$-X_5-32X_4-32^2X_3-32^3X_2-32^4X_1=[PARAMETER]$$

I want to get all the solutions to this equation with a given parameter(integer) and all of the solutions has to be integers $(X_1,X_2,X_3,X_4,X_5)$.

Is it possible? If so how?

Thanks.

Answer 1

Set $X_1$, $X_2$, $X_3$, $X_4$ to be whatever you like, and then set

$$X_5=-[PARAMETER]-32X_4-32^2X_3-32^3X_2-32^4X_1$$

This gives you all possible solutions because given $X_1$, $X_2$, $X_3$, $X_4$, then $X_5$ can be at most one value (the solution to the above equation), and this value is valid as the above equation yields an integer.