If m and n are positive integers and m*n = 40 what is m+n?

\(\left. \begin{gathered}\\
m,n\,\, \geqslant 1\,\,\,{\text{ints}}\,\, \hfill \\\\
mn = 40\, \hfill \\ \\
\end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,m\,\,{\text{and}}\,\,n\,\,{\text{are}}\,\,{\text{pairs}}\,\,{\text{of}}\,{\text{positive}}\,\,{\text{factors}}\,\,{\text{of}}\,\,40\)

\(? = m + n\)

This is a perfect opportunity to present our "T diagram" (see image attached), GMATH´s creation to help students find ALL positive factors of a given positive integer.

We all know that 40 = (2^3)*5 has 4*2 = 8 positive factors. The T technique shows them explicitly and, more than that, in the corresponding pairs (in the four rows)!

We have put the number of positive factors of each positive factor of 40 in parentheses. A quick inspection in each pair of divisors shows us that:

\(\left( 1 \right)\,\,\,\, \Rightarrow \,\,\,\left( {m,n} \right) = \left( {{2^3},5} \right)\,\,\,\,\, \Rightarrow \,\,\,? = 13\,\,\,\,\, \Rightarrow \,\,\,{\text{SUFF}}.\,\,\,\,\,\,\)

\(\left( 2 \right)\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}\\
\,\left( {m,n} \right) = \left( {2 \cdot 5,{2^2}} \right)\,\,\,\,\, \Rightarrow \,\,\,? = 14\,\,\,\,\, \hfill \\\\
\,\left( {m,n} \right) = \left( {{2^3},5} \right)\,\,\,\,\, \Rightarrow \,\,\,? = 13\,\,\,\, \hfill \\ \\
\end{gathered} \right. \Rightarrow \,\,\,{\text{INSUFF}}.\)


The correct answer is therefore (A).


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.