BTGmoderatorDC wrote:If x is a positive integer, is x a prime integer?
(1) x+1 is a prime number.
(2) x-5 is a prime number.
Source: GMAT Prep
$$x \geqslant 1\,\,\operatorname{int} $$
$$x\,\,\mathop = \limits^? \,\,{\text{prime}}$$
$$\left( 1 \right)\,\,x + 1\,\,{\rm{prime}}\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,x = 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,x - 5\,\,{\rm{prime}}\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 7\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,x = 8\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\left\{ \matrix{
\,\left( 2 \right)\,\,\, \Rightarrow \,\,\,x - 5 \ge 2\,\,\, \Rightarrow \,\,\,x \ge 7\,\,\,\left( * \right) \hfill \cr
\,\left( 1 \right)\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x + 1\,\,{\rm{odd}}\,\, \ge \,\,{\rm{9}}\,\,\,\,\, \Rightarrow \,\,\,\,\,x \ge 8\,\,{\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
In words: x+1 is prime and greater than 7, hence x+1 is odd. Therefore x is even and greater than 6, hence x is not prime!
The correct answer is (C).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
