adkikani
Is n an integer greater than 14?
(1) \(3n\) is a positive integer.
(2) \(\frac{n}{3}\) is a positive integer.
Important: the question asked is DIFFERENT from this one: "If n is an integer, is it greater than 14?"
More explicitly: If n is not an integer (this IS possible), then we have an example of an answer to the question asked in the negative!
\(?\,\,\,:\,\,n\,\,{\rm{is}}\,\,{\rm{an}}\,\,{\rm{integer}}\,\,{\rm{greater}}\,\,{\rm{than}}\,\,14\)
\(\left( 1 \right)\,\,3n \ge 1\,\,{\mathop{\rm int}} \,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,n = {1 \over 3}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,n = 15\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,{n \over 3} \ge 1\,\,{\mathop{\rm int}} \,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,n = 3\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,n = 15\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,\left\{ \matrix{\\
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,n = 3\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \\
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,n = 15\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{E}} \right)\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.