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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## W, X, Y and Z are four different positive integers. tagged by: Brent@GMATPrepNow ##### This topic has 2 expert replies and 0 member replies ### GMAT/MBA Expert ## W, X, Y and Z are four different positive integers. ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult 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 Difficulty level: 650 - 700 Source: www.gmatprepnow.com _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use our video course along with Sign up for our free Question of the Day emails And check out all of our free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 Brent@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 Difficulty level: 650 - 700 Source: www.gmatprepnow.com Beautiful problem, Brent. Congrats! $$W,X,Y,Z\,\,\, \ge 1\,\,\,{\rm{different}}\,\,{\rm{ints}}\,\,\left( * \right)$$ $$\left\{ \matrix{ X = ZY + W\,\,,\,\,\,W < Y\,\,\,\left( {**} \right) \hfill \cr \left( * \right)\,\,\,1 \le W < Y\,\,\, \Rightarrow \,\,\,Y \ge 2\,\,\,\left( {***} \right) \hfill \cr} \right.\,$$ $$? = Z$$ $$\left( 1 \right)\,\,X = W + 4\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,W + 4 = ZY + W\,\,\,\, \Rightarrow \,\,\,\,\,ZY = 4\,\,\,\,\mathop \Rightarrow \limits_{\left( {***} \right)}^{\left( * \right)} \,\,\,\,\,\,\left( {Y,Z} \right) = \left( {4,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$ [The viability of the unique (Y,Z)=(4,1) is an examinerÂ´s burden. In this case it is nice to mention 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) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12827 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 Brent@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 Difficulty level: 650 - 700 Source: www.gmatprepnow.com Target question: What is the value of Z? Given: When X is divided by Y, the quotient is Z and the remainder is W. ------ASIDE------- There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R" For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2 Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3 --------------------------------------- So, from the given information, we can write: X = YZ + W Statement 1: W = X - 4 Take X = YZ + W and replace X with X - 4 to get: X = YZ + X - 4 Subtract X from both sides: 0 = YZ - 4 Rewrite as: 4 = YZ We're told that Y and Z are DIFFERENT positive INTEGERS. So, there are only 2 possible cases: case i: Y = 1 and Z = 4 case ii: Y = 4 and Z = 1 case i yields a CONTRADICTION. If Y = 1, then we are dividing X by 1, and if we divide by 1, the remainder will always be ZERO. In other words, if Y = 1, then W = 0 However, we are told that W is a POSITIVE integer. So, we can definitely rule out case i, which means it MUST be the case that Y = 4 and Z = 1 (case ii) So, the answer to the target question is Z = 1 Since we can answer the target question with certainty, statement 1 is SUFFICIENT Statement 2: W + Z = 4 Since W and Z are DIFFERENT positive INTEGERS. So, there are only 2 possible cases: case i: W = 1 and Z = 3 case ii: W = 3 and Z = 1 Let's check each case to see whether each case yields a contradiction. case i: W = 1 and Z = 3 We get: "When X is divided by Y, the quotient is 3 and the remainder is 1" So, for example, it could be the case that X = 7 and Y = 2 We get: When 7 is divided by 2, the quotient is 3 and the remainder is 1 So, case i is possible. (In this case, the answer to the target question is Z = 3 case ii: W = 3 and Z = 1 We get: "When X is divided by Y, the quotient is 1 and the remainder is 3" So, for example, it could be the case that X = 7 and Y = 4 We get: When 7 is divided by 4, the quotient is 1 and the remainder is 3 So, case ii is also possible. (In this case, the answer to the target question is Z = 1 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT Answer: A Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use our video course along with Sign up for our free Question of the Day emails And check out all of our free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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