On $\mathbb{N}\times\mathbb{N}$ define the relation $R$, where $((a,b),(c,d)) \in R$ if and only if $\gcd(a,b)=\gcd(c,d)$. Show that $R$ is a equivalence relation.
Reflexivity :
For every $(a,b) \in \mathbb{N}\times\mathbb{N}$ we have $\gcd(a,b)=\gcd(b,a)$ and $\gcd(c,d)= \gcd(d,c)$
so $(a,b)R(a,b)$ and $R$ is reflexive
Symmetry :
Therefore $(c,d)R (d,c)$
Transitivity :
$\gcd(a,b)=x, \gcd(c,d)=y$ AND $\gcd(e,f)=z$. $\gcd(a,b)= \gcd(c,d)$ and $\gcd(c,d)= \gcd(e,f)$ therefore $\gcd(a,b)=\gcd(e,f)$.
I am confused with the transitivity. Is that right ?