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# State/Solutions to homogeneous linear Diophantine equations can be found with the LCM

Proposition: (Solutions to homogeneous linear Diophantine equations can be found with the LCM) Suppose that $a,b \in \ZZ$, and $a,b \neq 0$. Let $\ell = LCM(a,b)$ be their least common multiple. Let $S$ be the set of solutions to the homogeneous linear Diophantine equation $ax + by = 0$: $$S = \{ (x,y) \in \ZZ^2 \mbox{ such that } ax + by = 0 \}.$$

For all $n \in \ZZ$, define: $$s(n) = \left( \frac{\ell n}{a}, -\frac{\ell n}{b} \right) .$$ Then, $s$ is a bijection from $\ZZ$ to $S$.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Least common multiple, Def/Bijection

The following statements and definitions rely on the material of this page: Def/Domain topograph

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