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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 ## Challenge question: j and k are positive integers tagged by: Brent@GMATPrepNow ##### This topic has 2 expert replies and 0 member replies ### GMAT/MBA Expert ## Challenge question: j and k are positive integers ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult j and k are positive integers, and n = 10^j + k. Is n divisible by 15? (1) j and k are each divisible by 3 (2) j and k are each divisible by 5 Answer: A Difficulty level: 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: j and k are positive integers, and n = 10^j + k. Is n divisible by 15? (1) j and k are each divisible by 3 (2) j and k are each divisible by 5 Source: www.gmatprepnow.com $$n = {\rm{1}}{{\rm{0}}^{\rm{j}}} + k\,\,\,\,;\,\,\,\,\,j,k\,\,\, \ge 1\,\,\,\,{\rm{ints}}\,\,\,\left( * \right)$$ $${n \over {3 \cdot 5}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}}$$ $$\left( 1 \right)\,\, \cap \,\,\,\left( * \right)\,\,\, \Rightarrow \,\,\,\left\{ \matrix{ j = 3M,\,\,M \ge 1\,\,{\mathop{\rm int}} \hfill \cr k = 3L,\,\,L \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.$$ $$n\,\,\, = \underbrace {{{\left( {{{10}^{M\, \ge \,1}}} \right)}^3}}_{\left\langle {100 \ldots 0} \right\rangle \,\,{\rm{not}}\,\,{\rm{div}}\,\,{\rm{by}}\,\,3} + \underbrace {3L}_{{\rm{div}}\,\,{\rm{by}}\,\,3}\,\, = \,\,{\rm{not}}\,\,{\rm{div}}\,\,{\rm{by}}\,\,\,3\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle$$ $$\left( 2 \right)\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {j,k} \right) = \left( {5,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,n = 100005\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {\sum\nolimits_{{\rm{digits}}} {\,\,{\rm{and}}\,\,{\rm{final}}\,\,{\rm{digit}}} } \right) \hfill \cr \,{\rm{Take}}\,\,\left( {j,k} \right) = \left( {5,10} \right)\,\,\,\,\,\, \Rightarrow \,\,\,n = 100010\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\left( {\sum\nolimits_{{\rm{digits}}} {} } \right) \hfill \cr} \right.$$ This solution follows 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: 12836 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 Brent@GMATPrepNow wrote: j and k are positive integers, and n = 10^j + k. Is n divisible by 15? (1) j and k are each divisible by 3 (2) j and k are each divisible by 5 Given: n = 10^j + k Target question: Is n divisible by 15? Key concepts: - If n is divisible by 15, then n must be divisible by 3 AND by 5 - If n is divisible by 3, then the sum of n's digits must be divisible by 3 (for example, we know that 747 is divisible by 3, because 7+4+7 = 18, and 18 is divisible by 3) Statement 1: j and k are each divisible by 3 First recognize that 10^j = 1 followed by j zeros. For example, 10^15 = 1,000,000,000,000,000 (1 followed by 15 zeros) Next, recognize that, if k is divisible by 3, then the sum of n's digits must be divisible by 3 So, if n = 10^j + k, then the sum of n's digits will be 1 greater than some multiple of 3. [since we're adding 1 and several zeros to a number that is divisible by 3] For example, if j = 6 and k = 24, then n = 10^j + k = 10^6 + 24 = 1,000,024. In this case, the sum of n's digits = 1+0+0+0+0+2+4 = 7, which is 1 greater than a multiple of 3. Likewise, if j = 9 and k = 75, then n = 10^j + k = 10^9 + 75 = 1,000, 000,075. In this case, the sum of n's digits = 13, which is 1 greater than a multiple of 3. And, if j = 15 and k = 99, then n = 10^15 + 99 = 1,000, 000,000,000,099. In this case, the sum of n's digits = 19, which is 1 greater than a multiple of 3. And so on. Since the sum of n's digits will always be 1 greater than some multiple of 3, we can be certain that n is NOT divisible by 3 So, by the above property, n is NOT divisible by 15 Since we can answer the target question with certainty, statement 1 is SUFFICIENT Statement 2: j and k are each divisible by 5 There are several values of j and k that satisfy statement 2. Here are two: Case a: j = 5 and k = 5. In this case, n = 10^j + k = 10^5 + 5 = 100,005. Since 100,005 is divisible by 3 and by 5, the answer to the target question is YES, n IS divisible by 15 Case b: j = 5 and k = 10. In this case, n = 10^j + k = 10^5 + 10 = 100,010. Since 100,010 is NOT divisible by 3, the answer to the target question is NO, n is NOT divisible by 15 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! • Free Practice Test & Review How would you score if you took the GMAT 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 • 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 • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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