Bunuel wrote:

m is a multiple of 13. Is mn a multiple of 195?

(1) n has every factor that 45 has.

(2) m is divisible by 18.

\(m = 13K,\,\,K\,\,\operatorname{int} \,\,\,\,\,\left( * \right)\)

\(\frac{{m \cdot n}}{{3 \cdot 5 \cdot 13}}\,\,\mathop = \limits^? \,\,\operatorname{int} \,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,\boxed{\,\,\frac{{m \cdot n}}{{3 \cdot 5}}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int} \,\,}\)

\(\left( 1 \right)\,\,\,n\,\,{\text{has}}\,\,{\text{5}}\,\,{\text{and}}\,\,{\text{3}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)

\(\left( 2 \right)\,\,\,{m \over {2 \cdot {3^2}}} = {\mathop{\rm int}} \,\,\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2 \cdot {3^2} \cdot 13,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)

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

Regards,

Fabio.

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