fskilnik wrote:
Is x/11 an integer?
(1) 5x/11 is an integer
(2) 7x/11 is an integer
\(\frac{x}{{11}}\,\,\mathop = \limits^? \,\,\operatorname{int}\)
\(\left( 1 \right)\,\,\,\frac{{5x}}{{11}}\,\, = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = \frac{{11}}{5}\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,\frac{1}{5}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)
\(\left( 2 \right)\,\,\,\frac{{7x}}{{11}}\,\, = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = \frac{{11}}{7}\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,\frac{1}{7}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)
\(\left( {1 + 2} \right)\,\,\,\,\frac{x}{{11}}\,\,\, = \,\,\,\frac{{15x}}{{11}} - \frac{{14x}}{{11}}\,\,\, = \,\,\,3 \cdot \left( {\frac{{5x}}{{11}}} \right) - 2 \cdot \left( {\frac{{7x}}{{11}}} \right)\,\,\, = \,\,\,3 \cdot \operatorname{int} - 2 \cdot \operatorname{int} \,\,\, = \,\,\,\operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)
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