Given $a$, $b$, and $c$ are integers and not equal to zero, any mathematical operations or problem constraints involving $a$, $b$, and $c$ will assume these restrictions. Let's work through the problem systematically. Could you clarify if you have specific conditions, equations, or constraints involving $a$, $b$, and $c$? For example:

• Are you solving an equation like $ax + by = c$?
• Are $a$, $b$, and $c$ coefficients in a quadratic or cubic equation?
• Are they part of a set or system of inequalities?

This context is crucial for proceeding accurately. Please provide the detailed problem or question involving these integers.

Here’s how I can help based on common scenarios:

1. Diophantine Equations: Solve for integer solutions.
2. Inequalities: Analyze ranges or relationships.
3. Modulo Arithmetic: Handle divisibility or modular constraints.
4. Polynomials: Work with integer coefficients.
5. Optimization: Maximize or minimize a function under given integer constraints.

Let me know the specifics to proceed!