If gcd of two positive integers is 2 then both integers are even

Question

The statement feels very obvious when you think about it, but I’m very stuck trying to prove it.

I first tried to prove it by contradiction but I didn’t get anywhere by doing that. I’ve been told that the best way to do it is to just use an example.

Surely there must be a better way of proving it mathematically? I just can’t seem to put my finger on it.

What is the definition of "evenness"? If it is being a multiple of $2$, then having $2$ as a 'divisor' proves it — user340297, Oct 30 '17 at 12:47
$\text{GCD}(x,y)=d$ means that $x=dp;;y=dq$ for some integer $p,q$. Now if you substitute $d=2$ you have your proof — Raffaele, Oct 30 '17 at 12:48

score 1 · Answer 1 · edited Oct 30 '17 at 13:09

1

Observation: $gcd(a,b)|a$

This is a direct consequence of the fact that the gcd is a divisor of $a$. Therefore if the gcd is $2$, then $2|a$ so $a$ is even. Similarly, $b$ is even.

The reverse isn’t true. $4$ and $8$ are even but their $gcd$ is $4$, not $2$. However, there’s a simple way to patch the theorem to make it an if and only if. Do you see what it is?

edited Oct 30 '17 at 13:09

Oscar Lanzi

39,403

answered Oct 30 '17 at 12:50

Stella Biderman

31,155

Two positive even integers have a gcd of 2 if and only if the difference of the two integers is 2? – Oct 30 '17 at 13:44
@Thestrugglingstudent That's a good observation, though it's not quite always true. $gcd(2,2)=gcd(-6,2)=2$, but in both cases the difference isn't 2. I was thinking that two numbers are both even if their gcd is a multiple of $2$. – Stella Biderman Oct 30 '17 at 14:17
Ah, I see. Thank you very much! – Oct 30 '17 at 14:58

If gcd of two positive integers is 2 then both integers are even

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