Intersection of Additive Groups of Integer Multiples

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Theorem

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

Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding additive groups of integer multiples.


Then:

$\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\lcm \set {m, n} \Z, +}$


Proof

By definition:

$m \Z = \set {x \in \Z: m \divides x}$

Thus:

\(\displaystyle m \Z \cap n \Z\) \(=\) \(\displaystyle \set {x \in \Z: n \divides x} \cap \set {x \in \Z: m \divides x}\)
\(\displaystyle \) \(=\) \(\displaystyle \set {x \in \Z: n \divides x \land m \divides x}\)
\(\displaystyle \) \(=\) \(\displaystyle \set {x \in \Z: \lcm \set {m, n} \divides x}\)
\(\displaystyle \) \(=\) \(\displaystyle \lcm \set {m, n} \Z\)

Hence the result.

$\blacksquare$


Sources