Let's say I have 3 equations: \begin{align*} 75a + 5 = x\\ 101b + 2 = x\\ 163c + 3 = x \end{align*} How do I proceed to find the smallest $x$ where $x \geq 2000$ and $a,b,c$ are Natural numbers? Would that method work for $n$ equations?
Alternatively (if this is a bad question) what topic of math would help me solve this equation?
The result should be 237005.
(unfortunately I have no idea what topic of math this question belongs to so apologies for bad tags)
On
Your system can be written as \begin{align} & x \equiv5 \text{ mod } 75 \\ & x \equiv 2 \text{ mod } 101 \\ & x \equiv 3 \text{ mod }163 \end{align}
This is a system of congruences and can be solved using this: https://en.wikipedia.org/wiki/Chinese_remainder_theorem - see paragraph on constructive proof.
This is an example of the Chinese Remainder Theorem. It says you can find a solution as long as the multipliers of $a,b,c$ are coprime. The solutions will recur at intervals of $75 \cdot 101 \cdot 163$ The Wikipedia page gives several approaches to find a solution.