Subgroup of Additive Group of Integers Generated by Two Integers

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Theorem

Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\gen {m, n}$ be the subgroup of $\struct {\Z, +}$ generated by $m$ and $n$.


Then:

$\gen {m, n} = \struct {\gcd \set {m, n} \Z, +}$

That is, the additive groups of integer multiples of $\gcd \set {m, n}$, where $\gcd \set {m, n}$ is the greatest common divisor of $m$ and $n$.


Proof

By definition:

$\gen {m, n} = \set {x \in \Z: \gcd \set {m, n} \divides x}$


Hence the result.

$\blacksquare$


Sources