Is this a correct solution?

Show that if $n\in\mathbb{Z}$ then $(10n+3,5n+2)=1$. $(a,b)$ being the greatest common divisor.

We can use the Eucledean algorithm to get
$$
10n+3=1\cdot(5n+2)+5n+1
$$
$$
5n+2=1\cdot(5n+1)+1
$$
$$
5n+1=(5n+1)\cdot 1+0
$$
and we see that the last nonzero remainder is $1$ therefore $(10n+3,5n+2)=1$.
Is there a more elegant solution?

Thank you!