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Imagine trying to prove a claim that is too chaotic to solve with a single, sweeping algebraic formula.
Instead of hunting for one master equation, what if you could carve the entire universe of possibilities into a few neat boxes and verify each one individually?

The Strategy of Exhaustion

A proof by exhaustion (or proof by cases) proves a statement by breaking the entire domain into a finite set of sub-situations and proving the statement holds true in every single one.
For the proof to be mathematically valid, your cases must be exhaustive (covering every possible scenario) and ideally mutually exclusive (having no ambiguous overlap).
In 1976, mathematicians Kenneth Appel and Wolfgang Haken famously used this method with a supercomputer to prove the Four Color Theorem by exhausting all 1,936 distinct map configurations.
How do we decide which cases to split an algebra problem into?

Modular Partitions in Number Theory

The Division Algorithm guarantees that when you divide any integer nn by an integer dd, the remainder must be one of 0,1,2,…,d−10, 1, 2, \dots, d-1.
This gives us a natural, complete case split: every integer nn can be uniquely written as n=dq+rn = dq + r, where r∈{0,1,…,d−1}r \in \{0, 1, \dots, d-1\}.
Let's prove that for every integer nn, the expression n2+2n^2 + 2 is never divisible by 44.