Bunuel wrote:

Is x^2 + 9 prime?

(1) x is odd

(2) 3≤x≤7

\({x^2} + 9\,\,\,\mathop = \limits^? \,\,\,{\rm{prime}}\)

\(\left( 1 \right)\,\,\,x\,\,{\rm{odd}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x^2} + 9\,\, \ge \,\,\,10\,\,\,{\rm{and}}\,\,{\rm{even}}\,\,\,\,\,\left( {{\rm{also}}\,\,{\rm{when}}\,\,x \le - 1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\)

\(\left( 2 \right)\,\,\,3 \le x \le 7\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,x = 3\,\,\,\, \Rightarrow \,\,\,{x^2} + 9 = 18\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr

\,{\rm{Take}}\,\,x = \,\,\sqrt {10} \,\,\,\left( {\sqrt 9 < \,\,\sqrt {10} < \sqrt {49} } \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x^2} + 9 = 19\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\)

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

Regards,

Fabio.

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)

Our high-level "quant" preparation starts here: https://gmath.net