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.

_________________

Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)

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