How to prove an equivalence relation of (a,b) | a and b have a common factor grater than 1?

Question

Let $R$ be the following relation on the set of positive integers greater than one.

R = {(a,b) | a and b have a common factor greater than one}

Is $R$ an equivalence relation?

So far I have:

Reflexive

$gcd(a,b) > 1$

$gcd(a,a) = a > 1$

therefore $(a,a)∈R$

I believe this is enough proof for reflexive but I'm not sure how to prove symmetry or transitivity.

Answer 1

You proved correctly that $R$ is reflexive. It is symmetric, because if $d>1$ and $d$ is a common factor of $a$ and $b$, then $d$ is also a factor of $b$ and $a$. But $R$ is not transitive, since $(2,6)\in R$ and $(6,3)\in R$, but $(2,3)\notin R$. Therefore, $R$ is not an equivalence relation.