lmuenzel wrote:
How many even integers N exist such that x < N < x + 10?
(1) x is not odd
(2) x is not an integer
\(x\,\,\, < \,\,\,N\,\,{\text{even}}\,\,\, < \,\,\,x + 10\)
\(? = \# N\)
\(\left( 1 \right)\,\,x \ne {\text{odd}}\,\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,x{\text{ = 0}}\,\,\,\, \Rightarrow \,\,\,\,? = 4\,\,\,\,\,\,\,\left[ {2,4,6\,\,{\text{and}}\,\,8} \right] \hfill \\\\
\,{\text{Take}}\,\,x = 0.1\,\,\,\, \Rightarrow \,\,\,\,? = 5\,\,\,\,\,\,\,\left[ {2,4,6,8\,\,{\text{and}}\,\,10} \right] \hfill \\ \\
\end{gathered} \right.\)
\(\left( 2 \right)\,\,x \ne \operatorname{int} \,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x < \left\langle x \right\rangle \leqslant N \leqslant \left\langle {x + 9} \right\rangle < x + 10\)
\(\Rightarrow \,\,\,\,\left\{ \begin{gathered}\\
\,\left\langle x \right\rangle \,\,{\text{odd}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {x + 9} \right\rangle \,\,{\text{even}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}} = {\text{5}}\,\,\,\left[ {\left\langle {x + j} \right\rangle :j \in \left\{ {1,3,5,7,9} \right\}} \right] \hfill \\\\
\,\left\langle x \right\rangle \,\,{\text{even}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {x + 9} \right\rangle \,\,{\text{odd}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}} = {\text{5}}\,\,\,\left[ {\left\langle {x + j} \right\rangle :j \in \left\{ {0,2,4,6,8} \right\}} \right] \hfill \\ \\
\end{gathered} \right.\,\,\,\, \Rightarrow \,\,\,\,\,? = 5\)
\(\left( * \right)\,\,\left\langle r \right\rangle \,\, = \,\,{\text{smallest}}\,\,{\text{integer}}\,\,{\text{greater}}\,\,{\text{than}}\,\,r\)
The correct answer is therefore (B).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.