# Finding the second factor by employing symmetry

The factors of $$72$$ are $$1,2,3,4,6,8,9,12,18,24,36,72$$. The factors can be organized as such:

• $$m=1,2,3,4,6,8$$, and
• $$n=72,36,24,18,12,9$$.

Knowing the symmetric properties and that the $$12$$ factors exist. Is there a way to rewrite $$n$$ as a function of $$m$$ without actually diving $$72$$ by $$m$$? I tried adding and subtracting $$m$$ and $$n$$ to find a pattern as to get a clue as to what value the other might be without actually dividing.

• The defining relation between $m$ and $n$ is $mn=72$. There is no polynomial expression of $m$ in terms of $n$ nor vice-versa because division cannot be done by addition and multiplication. – lhf Sep 26 '13 at 16:16