Solution: Let $\phi : G \to \mathbb{Z} _{10}$ be $\psi : G \to \mathbb{Z} _{15}$ be surjective homomorphisms. From the first isomorphism theorem, \[ \frac{G}{\ker \phi } \cong \mathbb{Z} _{10} \quad \text{ and }\quad \frac{G}{\ker \psi } \cong \mathbb{Z} _{15}. \] Therefore, \[ \left\vert \frac{G}{\ker \phi } \right\vert = 10 \quad \text{ and } \quad \left\vert \frac{G}{\ker \psi } \right\vert = 15, \] which implies \begin{align*} \frac{\vert G \vert }{10} = \vert \ker \phi \vert \quad \text{ and } \quad \frac{\vert G \vert }{15} = \vert \ker \psi \vert. \end{align*} Hence, \[ 10 \mid \vert G \vert \text{ and } 15 \mid \vert G \vert \implies \mathrm{lcm} (10,15) = 30 \mid \vert G \vert \]

In general, if $\mathbb{Z} _m$ and $\mathbb{Z} _n$ are homomorphic images of $G$, then $\mathrm{lcm}(m,n) $ must divide the order of $G$. It can be further generalize as if $H_1$ and $H_2$ are two groups of order $m$ and $n$ are homomorphic images of $G$, then $\mathrm{lcm}(m,n) $ must divide order of $G$.