Existence of Latin Squares

Theorem
For each $n \in \Z_{>0}$ there exists at least one Latin square of order $n$.

Proof
The Cayley table of a finite group of order $n$ is a Latin square, from Group has Latin Square Property.

For every $n \in \Z_{>0}$ there exists a cyclic group of order $n$.

It follows that for every $n \in \Z_{>0}$ the Cayley table of the cyclic group of order $n$ is a Latin square of order $n$.