MathRevolution wrote:
[
Math Revolution GMAT math practice question]
If the integers \(p, q, r\), and \(s\) satisfy \(p<q<r<s\), are they consecutive?
\(1) r-q=1\)
\(2) (p-q)(r-s)=1\)
Beautiful problem, Max. Congrats!
\(p < q < r < s\,\,\,\,{\rm{ints}}\,\,\,\,\,\left( * \right)\)
\({\text{?}}\,\,{\text{:}}\,\,\,{\text{consecutives}}\,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\boxed{\,\,s - p\,\,\mathop { = \,}\limits^? \,3\,\,}\)
\(\left( 1 \right)\,\,\,\,r = q + 1\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,2,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,2,4} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,\left( {p - q} \right)\left( {r - s} \right) = 1\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,2,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,3,4} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,\,\,\left( {p - q} \right)\left( {r - s} \right) = 1\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,p - q\,\,{\rm{and}}\,\,r - s\,\,{\rm{are}}\,\,{\rm{divisors}}\,\,{\rm{of}}\,\,1\,\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,\left\{ \matrix{
\,\left( {\rm{I}} \right)\,\,\,\,\left( {p - q,r - s} \right) = \left( {1,1} \right) \hfill \cr
\,\,\,{\rm{OR}} \hfill \cr
\,\left( {{\rm{II}}} \right)\,\,\,\left( {p - q,r - s} \right) = \left( { - 1, - 1} \right) \hfill \cr} \right.\)
\(\left( {\rm{I}} \right)\,\,\,\,\left\{ \matrix{
p - q = 1 \hfill \cr
r - s = 1 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{impossible}}\,\,\,\,\,\,\,\left( {p < q\,\,\,{\rm{and}}\,\,\,r < s} \right)\)
\(\left( {{\rm{II}}} \right)\,\,\,\,\left\{ \matrix{
p - q = - 1 \hfill \cr
r - s = - 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\mathop \sim \limits^{\left( 1 \right)} \,\,\,\,\,\,\,\left\{ \matrix{
p - q = - 1 \hfill \cr
q + 1 - s = - 1 \hfill \cr} \right.\,\,\,\,\,\, \sim \,\,\,\,\,\left\{ \matrix{
p - q = - 1 \hfill \cr
q - s = - 2 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,p - s = - 3\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
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