Closed Algebraic Structure/Examples/2^m 3^n under Multiplication

Examples of Closed Algebraic Structures
Let $S$ be the set defined as:
 * $S := \set {2^m 3^n: m, n \in \Z}$

Then the algebraic structure $\struct {S, \times}$ is closed.

Proof
Let $a, b \in S$ such that:

where $m_1, m_2, n_1, n_2 \in \Z$.

Then: