BTGmoderatorDC wrote:If x and y are positive integers greater than 1 such that x - y and x/y are both even integers, which of the following numbers must be non-prime integers?
I. x
II. x + y
III. y/x
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II and III
Source: Manhattan Prep
Nice conceptual problem!
\[x,y\,\, \geqslant 2\,\,{\text{ints}}\,\,\,\,\left( * \right)\]
\[x - y\,\, = {\text{even}}\,\,\,\,\left( 1 \right)\]
\[\frac{x}{y}\,\, = \,\,{\text{even}}\,\,\,\,\left( {\mathop {\, \Rightarrow }\limits^{\left( * \right)} \,\,\,\,{\text{even}} \geqslant {\text{2}}} \right)\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\]
\[?\,\,\,:\,\,\,{\text{must}}\,\,{\text{be}}\,\,{\text{int}}\,\,\underline {{\text{not}}} \,\,{\text{primes}}\]
\[\left( 1 \right) \cap \left( * \right)\,\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}
\,x,y\,\,{\text{both}}\,\,{\text{odd}}\,\,\,\mathop \geqslant \limits^{\left( * \right)} \,\,\,\,3 \hfill \\
\,\,\,\,\,\,\,\,\,{\text{or}} \hfill \\
\,x,y\,\,{\text{both}}\,\,{\text{even}}\,\,\,\mathop \geqslant \limits^{\left( * \right)} \,\,\,\,2 \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right) \cap \left( * \right)\,\,\,\, \Rightarrow \,\,\,x\,\, = \,\,\underbrace {{\text{even}}}_{ \geqslant \,\,2} \cdot {\text{y}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\left\{ \begin{gathered}
x \geqslant 4\,\,\,{\text{even}} \hfill \\
x\, > \,y\, \hfill \\
\end{gathered} \right.\]
\[\left( 1 \right) \cap \left( 2 \right) \cap \left( * \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\boxed{\,\,x \geqslant 4\,\,\,,\,\,\,x > y\,\,\,,\,\,\,\,\,y \geqslant 2\,\,\,\,\,\,\,{\text{both}}\,\,\,{\text{even}}\,\,}\]
\[{\text{I}}.\,\,\,{\text{Yes}}:\,\,\,\,x\,\, \geqslant \,\,4\,\,{\text{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,x\,\,\,\operatorname{int} \,\,{\text{not}}\,\,{\text{prime}}\,\,\,\,\,\]
\[{\text{II}}.\,\,{\text{Yes}}:\,\,\,x + y\,\, \geqslant \,\,6\,\,{\text{and}}\,\,{\text{even}}\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,x + y\,\,\,\,\operatorname{int} \,\,\,{\text{not}}\,\,{\text{prime}}\,\]
\[{\text{III}}.\,\,{\text{No}}:\,\,\,x > y \geqslant 2 > 0\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,0 < \frac{y}{x} < 1\,\,\,\, \Rightarrow \,\,\,\,\,\frac{y}{x}\,\,\,{\text{not}}\,\,\operatorname{int} \]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
