MathRevolution wrote:

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

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