BTGmoderatorDC wrote:If z is a multiple of 9 and w is a multiple of 4, is zw a multiple of 126?

(1) z is a multiple of 21
(2) w is a multiple of 25

Source: Veritas Prep

\[\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,;\,\,\,\,\,\frac{w}{{{2^2}}} = \operatorname{int} \,\,\,\,\,\left( * \right)\]
\[\frac{{zw}}{{2 \cdot {3^2} \cdot 7}}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int} \]
\[\left( 1 \right)\,\,\,\,\left\{ \begin{gathered}
\,\frac{z}{{3 \cdot 7}} = \operatorname{int} \,\,\,\, \cap \,\,\,\,\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\frac{z}{{{3^2} \cdot 7}} = \operatorname{int} \, \hfill \\
\,\frac{w}{{{2^2}}} = \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\frac{w}{2} = \operatorname{int} \, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\]
\[\frac{{zw}}{{2 \cdot {3^2} \cdot 7}} = \left( {\frac{z}{{{3^2} \cdot 7}}} \right) \cdot \left( {\frac{w}{2}} \right)\,\,\, = \,\,\,\operatorname{int} \,\, \cdot \,\,\,\operatorname{int} \,\,\,\, = \operatorname{int} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( 2 \right)\,\,\,\,\frac{w}{{{5^2}}} = \operatorname{int} \,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\,\left( {z,w} \right) = \left( {0,0} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\,\left( {z,w} \right) = \left( {{3^2},{2^2} \cdot {5^2}} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{7}}\,\,{\text{missing}}} \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\,\,\,\,\,\]

The correct answer is therefore [spoiler]__(A)_____[/spoiler] .

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

Regards,
Fabio.