Odd Divisor of Even Number also divides its Half

Theorem
Let $a, b \in \Z$ be integers.

Let $a$ be odd and $b$ be even.

Let:
 * $a \divides b$

where $\divides$ denotes divisibility.

Then:
 * $a \divides \dfrac b 2$

Proof
By definition of an even number:
 * $\exists r \in \Z: b = 2 r$

By definition of an odd number:
 * $\exists s \in \Z: a = 2 s + 1$

Thus:
 * $a \divides 2 \iff a = \pm 1$

in which case from One Divides all Integers:
 * $a \divides \dfrac b 2$

We have that:
 * $a \divides 2 r$

and so by Euclid's Lemma:
 * $a \divides r = \dfrac b 2$