• NEW! FREE Beat The GMAT Quizzes
Hundreds of Questions Highly Detailed Reporting Expert Explanations
• 7 CATs FREE!
If you earn 100 Forum Points

Engage in the Beat The GMAT forums to earn
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 ## A number is called "mystic" if it can be expressed tagged by: fskilnik@GMATH ##### This topic has 1 expert reply and 0 member replies ### GMAT/MBA Expert ## A number is called "mystic" if it can be expressed ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult GMATH practice exercise (Quant Class 17) A number is called "mystic" if it can be expressed as the sum of at least three consecutive positive integers greater than 1, in which the larger addend is more than twice the smaller addend. Note that 25 (=3+4+5+6+7) and 44 (=2+3+4+5+6+7+8+9) are mystic numbers (7>2*3 and 9>2*2). Which of the following is true? I. 124 is mystic II. 125 is mystic III. 126 is mystic (A) I. only (B) II. only (C) III. only (D) Exactly two of them (E) None of them Answer: ____(D)__ P.S.: I have edited this post once: the previous word "parcel" was substituted by the proper word "addend". _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br Last edited by fskilnik@GMATH on Thu Feb 21, 2019 4:30 am; edited 1 time in total ### 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 17) A number is called "mystic" if it can be expressed as the sum of at least three consecutive positive integers greater than 1, in which the larger parcel is more than twice the smaller parcel. Note that 25 (=3+4+5+6+7) and 44 (=2+3+4+5+6+7+8+9) are mystic numbers (7>2*3 and 9>2*2). Which of the following is true? I. 124 is mystic II. 125 is mystic III. 126 is mystic (A) I. only (B) II. only (C) III. only (D) Exactly two of them (E) None of them $$?\,\,\,:\,\,\,M + \left( {M + 1} \right) + \left( {M + 2} \right) + \ldots + N$$ $$M \ge 2$$ $$N > 2M\,\,\,\,\,\left[ { \Rightarrow \,\,\,\,\,N > 4\,\,\,\,\,\mathop \Rightarrow \limits^{N\,\,{\mathop{\rm int}} } \,\,\,\,\,N \ge 5} \right]$$ $${{N\left( {N + 1} \right)} \over 2}\,\,\,\mathop = \limits^{{\rm{arithm}}{\rm{.}}\,{\rm{seq}}{\rm{.}}} \,\,\,\underbrace {1 + 2 + \ldots + \left( {M - 1} \right)}_{{{M\left( {M - 1} \right)} \over 2}\,\,\,\,\left[ {{\rm{arith}}{\rm{.}}\,{\rm{seq}}{\rm{.}}} \right]} + M + \left( {M + 1} \right) + \ldots + N\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {{N\left( {N + 1} \right)} \over 2} - {{M\left( {M - 1} \right)} \over 2}$$ $$2\,\, \cdot \,\,?\,\,\, = \,\,\,{N^2} + N - {M^2} + M = \left( {N - M} \right)\left( {N + M} \right) + N + M = \left( {N + M} \right)\left( {N - M + 1} \right)$$ $$\left\{ \matrix{ \,N - M + 1 \ge 3\,\,\,\,\left[ { \ge \,\,{\rm{3}}\,\,{\rm{parcels}}\,\,\left( {{\rm{stem}}} \right)\,\,{\rm{,}}\,\,{\rm{fingers}}\,\,{\rm{trick}}} \right] \hfill \cr \,N + M > N - M + 1\,\,\,\left[ {M > 0.5} \right] \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\min \left( {N + M,N - M + 1} \right) = N - M + 1 \ge 3\,\,\,\,\left( * \right)$$ $$\left( {**} \right)\,\,\left\{ \matrix{ \,N + M\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M = \underbrace {N + M}_{{\rm{even}}} - \underbrace {2M}_{{\rm{even}}}\,\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M + 1\,\,{\rm{odd}} \hfill \cr \,N + M\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M = \underbrace {N + M}_{{\rm{odd}}} - \underbrace {2M}_{{\rm{even}}}\,\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M + 1\,\,{\rm{even}} \hfill \cr} \right.$$ $$2\,\, \cdot \,\,?\,\,\, = \,\,\,\left( {N + M} \right)\left( {N - M + 1} \right)\,\,\,{\rm{is}}\,\,{\rm{given}}\,\,{\rm{by}}\,\,\,{\rm{even}} \cdot {\rm{odd}}\,\,\left( {**} \right),\,\,{\rm{where}}\,\,\left( * \right)\,\,\,{\rm{odd}} \ge {\rm{3}}\,{\rm{,}}\,\,{\rm{even}}\,\, \ge 4\,\,\,\,\,\left( {***} \right)$$ $${\rm{I}}{\rm{.}}\,\,\,124\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 124 = 248 = {2^3} \cdot 31\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\left\{ \matrix{ \,{\rm{odd}} = 31 \hfill \cr \,{\rm{even}} = {2^3} \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{ \,N + M = 31 \hfill \cr \,N - M + 1 = {2^3} \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\, \ldots \,\,\, \Rightarrow \,\,\,\left( {N,M} \right) = \left( {19,12} \right)$$ $$124 = 12 + 13 + \ldots + 18 + 19\,\,\,{\rm{but}}\,\,{\rm{impossible}}\,\,\,\left( {N < 2M} \right)$$ $${\rm{II}}{\rm{.}}\,\,\,125\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 125 = 250 = 2 \cdot {5^3}\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,{\rm{one}}\,\,{\rm{option}}\,\,:\,\,\left\{ \matrix{ \,{\rm{odd}} = {5^2} \hfill \cr \,{\rm{even}} = 2 \cdot 5 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{ \,N + M = 25 \hfill \cr \,N - M + 1 = 10 \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\, \ldots \,\,\, \Rightarrow \,\,\,\left( {N,M} \right) = \left( {17,8} \right)$$ $$125 = 8 + 9 + \ldots + 16 + 17\,\,\,{\rm{is}}\,\,{\rm{viable}}!\,\,\,\left( {N > 2M} \right)$$ $${\rm{III}}{\rm{.}}\,\,\,126\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 126 = 252 = {2^2} \cdot {3^2} \cdot 7\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,{\rm{one}}\,\,{\rm{option}}\,\,:\,\,\left\{ \matrix{ \,{\rm{odd}} = 3 \cdot 7 \hfill \cr \,{\rm{even}} = {2^2} \cdot 3 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\mathop \ldots \limits^{{\rm{Do}}\,\,{\rm{it}}!} \,\,\,\, \Rightarrow \,\,\,\,\left( {N,M} \right) = \left( {16,5} \right)$$ $$126 = 5 + 6 + \ldots + 15 + 16\,\,\,{\rm{is}}\,\,{\rm{viable}}!\,\,\,\left( {N > 2M} \right)$$ YES, this is a very hard problem, but I believe many "pieces" of this solution are VERY useful for candidates aiming really outstanding performances! 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 • Award-winning private GMAT tutoring Register now and save up to$200

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

• Get 300+ Practice Questions

Available with Beat the GMAT members only code

• 5-Day Free Trial
5-day free, full-access trial TTP Quant

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 GMAT Exam
Know how you'd score today for \$0

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 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

### Top First Responders*

1 Ian Stewart 41 first replies
2 Brent@GMATPrepNow 36 first replies
3 Scott@TargetTestPrep 33 first replies
4 Jay@ManhattanReview 30 first replies
5 GMATGuruNY 23 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

### Most Active Experts

1 Scott@TargetTestPrep

Target Test Prep

159 posts
2 Max@Math Revolution

Math Revolution

91 posts
3 Brent@GMATPrepNow

GMAT Prep Now Teacher

56 posts
4 Ian Stewart

GMATiX Teacher

50 posts
5 GMATGuruNY

The Princeton Review Teacher

35 posts
See More Top Beat The GMAT Experts