The Question: Prove that (1 2) cannot be written as the product of 3 disjoint cycles.
The Attempt: Suppose (1 2) has a cycle decomposition into 3 disjoint cycles $m_1, m_2$, and $m_3$. Then (1 2) = $m_1 m_2 m_3$ and the order of (1 2) should be the least common multiple of $m_1, m_2,$ and $ m_3$.
Where do I go from here?