[Math] Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$

Question

Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$

Hi everyone, I would like to know if my assumption is justified for answering the above question. Any constructive feedback is greatly appreciated, thanks!

Since $a$ is even and $b$ is odd, then 2 will not a common factor so it suffices to take the gcd of $a/2$ and $b$ which is the same as the gcd of $a$ and $b$.

Answer 1

Your claim is correct, but can you prove it rigorously? It's good practice when beginning to look at problems in number theory to prove even the most basic statements in your arguments. A sketch of one route to a proof is given below, but I'd recommend trying to show this on your own before looking at it.

If you haven't done so before, show that $b$ odd $\implies$ $2\nmid b$. Thus by definition $2 \nmid d$ for $d = (a,b)$. You also know $a = 2m$ for some integer $m$ and $a = qd$ for some integer $q$ (why?). You can conclude from the equation $2m = qd$ and $2\nmid d$ that $2\mid q$ (why?) and therefore $2 \mid q' := q/2 \in \mathbb{Z}$. Now notice $a/2 = q'd$. Conclude from this that $(a/2,b) = d$.

(Another good approach would be to use the Fundamental Theorem of Arithmetic.)