Choose the right gear

We are given a set of gears, one equation to be satisfied (see picture below) and a number of constraints.

Terms on the left (work and cutter) are given, while A, B, C and PG must be chosen from a set of 29 elements (it is a set in common among unknowns, so we are not allowed to use the same gear twice for two different variables, even though there are duplicates).

How do we find the right gears?

Switching from Solver to OpenSolver

I was solving a very simple model found in a forum: given a total cost in € (or $\$$, if you are reading from across the ocean) for a set of projects, and the hourly cost of 3 technician involved, find the integer number of hours each one should spend on it. The aim is minimizing the wasted time, that is, solve the following

$$
 \min \left| TC - \sum_{i=1}^{3} c_{i} k_{i} \right| \; \; s.t. k_{i} \in \mathbb{N}
$$

The problem was to be solved for a number of rows, with different costs.