Bunuel wrote:
If a and b are positive integers, is ab < 6?
(1) 1 < a + b < 7
(2) ab = a + b
\(a,b \geqslant 1\,\,{\text{ints}}\,\,\,\left( * \right)\)
\({\text{ab}}\,\,\mathop < \limits^? \,\,6\)
\(\left( 1 \right)\,\,1 < a + b < 7\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {2,3} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\)
\(\left( 2 \right)\,\,ab = a + b\,\,\,\, \Rightarrow \,\,\,\,a\left( {b - 1} \right) = b\,\,\,\,\left( {**} \right)\)
\(a = 1\,\,\,{\text{OR}}\,\,\,b = 1\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,{\text{impossible}}\)
\(a,b\,\, \ge {\rm{2}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\left\{ \matrix{\\
a\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,b\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)\,\,} \,\,\,\,\,a \le b\, \hfill \cr \\
b - 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,b\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)\,\,} \,\,\,\,\,b = 2\, \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.