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.

_________________

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

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