GMATPrepNow wrote:
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W.
What is the value of Z?
1) W = X – 4
2) W + Z = 4
Beautiful problem, Brent. Kudos!
\(W,X,Y,Z\,\,\, \ge 1\,\,\,{\rm{different}}\,\,{\rm{ints}}\,\,\left( * \right)\)
\(\left\{ \matrix{
X = ZY + W\,\,,\,\,\,W < Y\,\,\,\left( {**} \right) \hfill \cr
\left( * \right)\,\,\,1 \le W < Y\,\,\, \Rightarrow \,\,\,Y \ge 2\,\,\,\left( {***} \right) \hfill \cr} \right.\,\)
\(? = Z\)
\(\left( 1 \right)\,\,X = W + 4\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,W + 4 = ZY + W\,\,\,\, \Rightarrow \,\,\,\,\,ZY = 4\,\,\,\,\mathop \Rightarrow \limits_{\left( {***} \right)}^{\left( * \right)} \,\,\,\,\,\,\left( {Y,Z} \right) = \left( {4,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)
(The viability of the unique (Y,Z)=(4,1) is an examiner´s burden. In this case it is nice to mention, as cheetan2u did, that (X,Y,Z,W)=(7,4,1,3) is viable. The problem is perfect!)
\(\left( 2 \right)\,\,W + Z = 4\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {X,Y,Z,W} \right) = \left( {7,2,3,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 3\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {X,Y,Z,W} \right) = \left( {7,4,1,3} \right)\,\,\,\, \Rightarrow \,\,\,? = 1 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{INSUFF}}{\rm{.}}\)
The correct answer is (A).
We follow the notations and rationale taught in the
GMATH method.
Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net