How many even integers N exist such that x < N < x + 1

[BTGmoderatorDC]

How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

OA B

Source: Veritas Prep

[fskilnik@GMATH]

How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer
Source: Veritas Prep

\[x\,\,\, < \,\,\,N\,\,{\text{even}}\,\,\, < \,\,\,x + 10\]
\[\left( 1 \right)\,\,x \ne {\text{odd}}\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x{\text{ = 0}}\,\,\,\, \Rightarrow \,\,\,\,? = 4\,\,\,\,\,\,\,\left[ {2,4,6\,\,{\text{and}}\,\,8} \right] \hfill \\
\,{\text{Take}}\,\,x = 0.1\,\,\,\, \Rightarrow \,\,\,\,? = 5\,\,\,\,\,\,\,\left[ {2,4,6,8\,\,{\text{and}}\,\,10} \right] \hfill \\
\end{gathered} \right.\]

\[\left( 2 \right)\,\,x \ne \operatorname{int} \,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x < \left\langle x \right\rangle \leqslant N \leqslant \left\langle {x + 9} \right\rangle < x + 10\]
\[ \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,\left\langle x \right\rangle \,\,{\text{odd}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {x + 9} \right\rangle \,\,{\text{even}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}} = {\text{5}}\,\,\,\left[ {\left\langle {x + j} \right\rangle :j \in \left\{ {1,3,5,7,9} \right\}} \right] \hfill \\
\,\left\langle x \right\rangle \,\,{\text{even}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {x + 9} \right\rangle \,\,{\text{odd}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}} = {\text{5}}\,\,\,\left[ {\left\langle {x + j} \right\rangle :j \in \left\{ {0,2,4,6,8} \right\}} \right] \hfill \\
\end{gathered} \right.\,\,\,\, \Rightarrow \,\,\,\,\,? = 5\]
\[\left( * \right)\,\,\left\langle r \right\rangle \,\, = \,\,{\text{smallest}}\,\,{\text{integer}}\,\,{\text{greater}}\,\,{\text{than}}\,\,r\]

The correct answer is therefore (B).


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

Regards,
Fabio.