Is n an integer greater than 14?

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.

Target question: Is n an integer greater than 14?

Statement 1: \(3n\) is a positive integer.
There are several values of n that satisfy statement 1. Here are two:
Case a: n = 15. In this case, the answer to the target question is YES, n is greater than 14
Case b: n = 3. In this case, the answer to the target question is NO, n is not greater than 14
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: \(\frac{n}{3}\) is a positive integer.
Important: Notice the SAME n-values that's satisfied statement 1 also satisfy statement 2. So let's reuse them.
Case a: n = 15. In this case, the answer to the target question is YES, n is greater than 14
Case b: n = 3. In this case, the answer to the target question is NO, n is not greater than 14
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Since we were able to use the same counter-examples to show that statements 1 and 2 alone are insufficient, the same counter-examples can be used to show that the combined statements are insufficient
That is...
Case a: n = 15. In this case, the answer to the target question is YES, n is greater than 14
Case b: n = 3. In this case, the answer to the target question is NO, n is not greater than 14
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent­

(1) n = 1, 2, 3 or 15, 3n is a positive integer. Insufficient.

(2) n = 3, 6, 9 or 15, n/3 is a positive integer. Insufficient.

Combining the two we get:
n = 3, 6, 9, 12 or 15, 3n and n/3 are both positive integers. Insufficient.

Hence, E.­