tensor product of division algebras

Let $$\Delta_{1}$$ and $$\Delta_{2}$$ be finite dimensional division algebras over field $$F$$ and $$\Delta_{1}$$ is central, then $$\Delta_{1}\otimes\Delta_{2} = M_{r}(E)$$ where $$E$$ is a division algebra, therefore we have $$\Delta_{1}^{op} \otimes \Delta_{1} \otimes \Delta_{2} \cong M_n(F) \otimes\Delta_{2} \cong M_r(E) \otimes \Delta_{1}^{op} \cong M_{r}(F) \otimes E \otimes\Delta_{1}^{op}\cong M_{r}(F) \otimes M_{q}(\Delta^{-}) \cong M_{rq}(F)\otimes\Delta^{-}$$ where n is the degree of $$\Delta_{1}$$ over $$F$$. this implies $$rq=n$$ and $$\Delta^{-} \cong\Delta_{2}$$, so $$r \mid[\Delta_{1} : F]$$.

MY QUESTION IS: is $$r\mid[\Delta_{2} : F]$$?

(so $$\Delta_{1}\otimes\Delta_{2}$$ will be division algebra if gcd($$[\Delta_{1} : F], [\Delta_{2} : F]$$) is 1)