Jazzmin wrote:

The product of the GCF and LCM of positive integers p and q is rs. What is the LCM of p and q?

1. \(r=\frac{400}{s}\)

2. The GCF of p and q is 20

\(\left\{ \matrix{

p,q\,\, \ge \,\,1\,\,\,{\rm{ints}} \hfill \cr

pq = GCF\left( {p,q} \right) \cdot LCM\left( {p,q} \right) = rs\,\,\,\left( * \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,? = LCM\left( {p,q} \right)\)

\(\left( 1 \right)\,\,\,rs = 400\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {1,400} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{400}}\,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {20,20} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{20}}\,\,\, \hfill \cr} \right.\)

\(\left( 2 \right)\,\,GCF\left( {p,q} \right) = 20\,\,\,\left\{ \matrix{

\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {p,q} \right) = \left( {20,20} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{20}}\,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {20,40} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{40}}\,\,\, \hfill \cr} \right.\,\)

\(\left( {1 + 2} \right)\,\,\,\,\,\left( * \right)\,\,\,\, \Rightarrow \,\,\,\,20 \cdot LCM\left( {p,q} \right) = 400\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = LCM\left( {p,q} \right)\,\,{\rm{ = }}\,\,{\rm{20}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)

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

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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