Lowest Common Multiple of Integers with Common Divisor

Theorem
Let $b, d \in \Z_{>0}$ be (strictly) positive integers

Then:
 * $\lcm \set {a b, a d} = a \lcm \set {b, d}$

where:
 * $a \in \Z_{>0}$
 * $\lcm \set {b, d}$ denotes the lowest common multiple of $m$ and $n$.

Proof
We have that:

Suppose $n \in \Z$ such that $a b \divides n$ and $a d \divides n$.

It will be shown that $a \lcm \set {b, d} \divides n$.

So:

Thus we have:


 * $a b \divides a \lcm \set {b, d} \land a d \divides a \lcm \set {b, d}$

and:
 * $a b \divides n \land a d \divides n \implies a \lcm \set {b, d} \divides n$

It follows from LCM iff Divides All Common Multiples that:
 * $\lcm \set {a b, a d} = a \lcm \set {b, d}$