# Existence of Latin Squares

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## 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 $n$.

$\blacksquare$