Topological space with fundamental group $\mathbb{Z}_n * \mathbb{Z}_m$

Question

So im doing an exercise that asks me to find a topological space that has a fundamental group isomorphic to $\mathbb{Z}_n * \mathbb{Z}_m$. I would like to know if the space that I found is correct or not. So we know that if we do the labeling of polygon of $n$ sides with the labeling scheme $a...a , n$ times by the Seifert Van-Kampen Theorem its fundamental group is isomorphic to $\mathbb{Z}_n$. So know we have found spaces that gives fundamental groups of $\mathbb{Z}_n$ for any $n$ natural. Now what I did was the wedge the spaces wich give the fundamental groups required, and since my spaces are the quotient of polygons I can always find a neighborhood $W_i$ in each one so that $p$ is a deformation retract of $W_i$ and so I can use the Seifert Van Kampen Theorem with some appropriate open sets that cover up the space and again I can conclude that the fundamental group of this spaces is isomorphic to what I want. Is this correct ?