Let $d_1$ and $d_2$ be two coprime integers. Show that for all $(u,v)\in\mathbb{Z}^2$, $$L_{\mathrm{prim}}(u,v,d_1 d_2) = L_{\mathrm{prim}}(u,v,d_1)\,L_{\mathrm{prim}}(u,v,d_2).$$
Let $d_1$ and $d_2$ be two coprime integers. Show that for all $(u,v)\in\mathbb{Z}^2$,
$$L_{\mathrm{prim}}(u,v,d_1 d_2) = L_{\mathrm{prim}}(u,v,d_1)\,L_{\mathrm{prim}}(u,v,d_2).$$