N=abc, N is a three digit number with hundreds digit a, tens digit b

\(N = \left\langle {abc} \right\rangle \,\,\,\,\,\left( {a \ne 0} \right)\)

\(a + b + c = 20\,\,\,\left( * \right)\,\,\,\)

\(? = b\)

\(\left( 1 \right)\,\,ac = 18\,\,\,\,\mathop \Rightarrow \limits^{{\rm{all}}\,\,{\rm{possibilities}}} \,\,\,\left\{ \matrix{\\
\,\left( {a,b,c} \right) = \left( {2,9,9} \right)\,\,\, \Rightarrow \,\,\,? = 9 \hfill \cr \\
\,\left( {a,b,c} \right) = \left( {9,9,2} \right)\,\,\, \Rightarrow \,\,\,? = 9 \hfill \cr \\
\,\left( {a,b,c} \right) = \left( {3,11,6} \right)\,\,\,{\rm{impossible}} \hfill \cr \\
\,\left( {a,b,c} \right) = \left( {6,11,3} \right)\,\,\,{\rm{impossible}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 9\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)

\(\left( 2 \right)\,\,a > b = c\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,a\,\,{\rm{even}}\,\,{\rm{ > }}\,\,\,{\rm{6}}\,\,\,\left( {a \le 6\,\,\,\, \Rightarrow \,\,\,b = c \ge 7} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {a,b,c} \right) = \left( {8,6,6} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)


The correct answer is therefore (D).

Obs.: in official questions, when the correct alternative choice is (D), we expect the unique answers (obtained in each statement alone) to be equal.


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

Regards,
Fabio.