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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 ## What is the value of the positive two-digit integer N? tagged by: fskilnik@GMATH ##### This topic has 1 expert reply and 0 member replies ### GMAT/MBA Expert ## What is the value of the positive two-digit integer N? ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult GMATH practice exercise (Quant Class 16) What is the value of the positive two-digit integer N? (1) N and the product of the digits of N are 12 units apart. (2) N is greater than the product of the digits of N. Answer: ____(E)__ _________________ 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 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 fskilnik@GMATH wrote: GMATH practice exercise (Quant Class 16) What is the value of the positive two-digit integer N? (1) N and the product of the digits of N are 12 units apart. (2) N is greater than the product of the digits of N. $$? = N = \left\langle {ab} \right\rangle \,\,\,\,\,\,\,\left[ {a \ge 1\,,\,b \ge 0\,\,\,{\rm{digits}}} \right]$$ $$\left( 1 \right)\,\,\,\left| {\left( {10a + b} \right) - ab} \right| = 12\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left| {\left( {b - 10} \right)\left( {1 - a} \right)} \right| = 2\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left( {10 - b} \right)\left( {a - 1} \right) = 2$$ $$\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b\,,\,\,a - 1} \right)\,\,{\rm{pair}}\,\,\,{\rm{of}}\,\,{\rm{positive}}\,\,{\rm{divisors}}\,\,{\rm{of}}\,\,2\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b,a - 1} \right) \in \left\{ {\left( {1,2} \right),\left( {2,1} \right)} \right\}$$ $$\left. \matrix{ \left\{ \matrix{ \,10 - b = 1 \hfill \cr \,a - 1 = 2 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {3,9} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 39\,\,{\rm{viable!}}\,\,\,\, \hfill \cr \left\{ \matrix{ \,10 - b = 2 \hfill \cr \,a - 1 = 1 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {2,8} \right)\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{viable!}} \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{or}}\,\,39$$ $$\left( * \right)\,\,\left( {10a + b} \right) - ab = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b - \underline {10} = 12 - \underline {10} \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {b - 10} \right)\left( {1 - a} \right) = 2$$ $$\left( {**} \right)\,\,\,\left\{ \matrix{ \,b - 10 < 0\,\,\,\, \Rightarrow \,\,\,\,\left| {b - 10} \right| = 10 - b \hfill \cr \,1 - a \le 0\,\,\,\, \Rightarrow \,\,\,\,\left| {1 - a} \right| = a - 1 \hfill \cr} \right.$$ $$\left( 2 \right)\,\,\,\left\{ \matrix{ \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 28\,\,\,\,\,\left[ {28 > 16} \right]\,\,\,\, \Rightarrow \,\,\,? = 28 \hfill \cr \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 39\,\,\,\,\,\left[ {39 > 27} \right]\,\,\,\, \Rightarrow \,\,\,? = 39 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{E}} \right)$$ We follow the notations and rationale taught in the GMATH method. Regards, Fabio. P.S.: we know 28 and 39 could be found by some "organized manual work"... but the importance of the solution is contained in the techniques presented, not in the numbers themselves! _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep 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 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 • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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