AbhishekRyu wrote:If a, b & c are integers, is abc odd?

(1) ab is odd
(2) bc is odd

\[a,b,c\,\,{\text{ints}}\,\,\,\left( * \right)\]
\[abc\,\,\mathop = \limits^? \,\,\,{\text{odd}}\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\boxed{\,\,?\,\,\,:\,\,\,a,b,c\,\,{\text{odd}}\,\,\,}\,\,\]

\[\left( 1 \right)\,\,\,ab = odd\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,a,b\,\,{\text{odd}}\,\,\,\,{\text{but}}\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,bc = odd\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,b,c\,\,{\text{odd}}\,\,\,\,{\text{but}}\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {0,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]

\[\left( {1 + 2} \right)\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]

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

Regards,
fskilnik.