BTGmoderatorDC wrote:If x and y are integers and 2 < x < y, does y = 16 ?
(1) The GCF of X and Y is 2.
(2) The LCM of X and Y is 48.
Source: Veritas Prep
\[2 < x < y\,\,{\text{ints}}\]
\[y\,\,\mathop = \limits^? \,\,16 = {2^4}\]
\[\left( 1 \right)\,\,GCF\left( {x,y} \right) = 2\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {{2^2},2 \cdot 3} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\text{No}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {2 \cdot {{3,2}^4}} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\text{Yes}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,LCM\left( {x,y} \right) = {2^4} \cdot 3\,\,\,\left\{ \begin{gathered}
\,\left( {\operatorname{Re} } \right){\text{Take}}\,\,\left( {x,y} \right) = \left( {2 \cdot {{3,2}^4}} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\text{Yes}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {{2^2}{{,2}^4} \cdot 3} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\text{No}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\left\{ \begin{gathered}
\,\left( 1 \right) \cap \left( 2 \right)\,\,\,\, \Rightarrow \,\,\,\,xy = GCF\left( {x,y} \right) \cdot LCM\left( {x,y} \right) = 3 \cdot {2^5}\,\,\,\left( * \right) \hfill \\
\,\left( 1 \right)\,\, \Rightarrow \left\{ \begin{gathered}
x = 2 \cdot M \hfill \\
y = 2 \cdot N \hfill \\
\end{gathered} \right.\,\,\,\,\,GCF\left( {M,N} \right) = 1\,\,,\,\,\left( * \right)\,\, \Rightarrow \,\,3 \cdot {2^3} = MN\,\,\,\,\left( {M,N \geqslant 2} \right) \hfill \\
\left( {M,N} \right) = \left( {{2^3},3} \right)\,\,\, \Rightarrow \,\,\,\left( {x,y} \right) = \left( {{2^4},2 \cdot 3} \right)\,\,\, \Rightarrow \,\,\,x > y\,\,{\text{impossible}} \hfill \\
\therefore \left( {M,N} \right) = \left( {{{3,2}^3}} \right)\,\,\, \Rightarrow \,\,\,\left( {x,y} \right) = \left( {2 \cdot {{3,2}^4}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \hfill \\
\end{gathered} \right.\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
