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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 largest value of non-negative integer N for tagged by: swerve ##### This topic has 3 expert replies and 1 member reply ### Top Member ## What is the largest value of non-negative integer N for ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult What is the largest value of non-negative integer N for which 10^N is a factor of 50!? A. 5 B. 6 C. 12 D. 15 E. 20 The OA is C Source: GMAT Prep ### GMAT/MBA Expert Elite Legendary Member Joined 23 Jun 2013 Posted: 10071 messages Followed by: 494 members Upvotes: 2867 GMAT Score: 800 Hi All, We're asked for the largest value of N for which 10^N is a factor of 50! This question comes down to 'prime factorization' and requires a degree of thoroughness on your part (to make sure that you find all of the 10s in 50!) To start, 50! = (50)(49)(48)(47)(46)(45).....(3)(2)(1), so 50! is a really big number. You do not have to calculate it though; you just have to find all of the 10s that are hidden in that product. 10 = (2)(5), so we should start by looking for all of the multiples of 5 in 50!... 5, 10, 15 and 20 each contain a "5" 25 actually contains TWO 5s (re: 5x5) 30, 35, 40 and 45 each contain a "5" 50 contains TWO 5s (re: 2x5x5) Thus, there are 4 + 2 + 4 + 2 = TWELVE 5s Each of those 5s can be multiplied by a 2 (and there are LOTS of 2s in 50!), so we'll end up with twelve 10s. Thus, the maximum possible value of N is 12. Final Answer: C GMAT assassins aren't born, they're made, Rich _________________ Contact Rich at Rich.C@empowergmat.com Newbie | Next Rank: 10 Posts Joined 14 Nov 2017 Posted: 6 messages swerve wrote: What is the largest value of non-negative integer N for which 10^N is a factor of 50!? A. 5 B. 6 C. 12 D. 15 E. 20 The OA is C Source: GMAT Prep Since 10=2*5, the goal is to count the number of 5's in 50! All multiples of 5 have a 5. There are 10 multiples of 5 from 1-50 inclusive, which you can find by using the counting formula: (Largest - Smallest)/Distance + 1 --> (50-5)/5 + 1 = 10. Since 25=5^2, there is an additional 5 in 25. Since 50 = 25*2, there is an additional 5 in 25. So 10+1+1 = 12. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1282 messages Followed by: 29 members Upvotes: 59 swerve wrote: What is the largest value of non-negative integer N for which 10^N is a factor of 50!? A. 5 B. 6 C. 12 D. 15 E. 20 Source: GMAT Prep $$N \geqslant 0\,\,\operatorname{int} \,\,{\text{such}}\,\,{\text{that}}\,\,\,\frac{{50!}}{{{{10}^N}}} = \operatorname{int} \,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\boxed{\,N \geqslant 0\,\,\operatorname{int} \,\,{\text{such}}\,\,{\text{that}}\,\,\,\frac{{50!}}{{{5^N}}} = \operatorname{int} \,\,}$$ $$\left( * \right)\,\,{\rm{5s}}\,\,{\rm{are}}\,\,{\rm{fewer}}\,\,{\rm{than}}\,\,{\rm{2s}}\,\,$$ $$? = N\max$$ $$? = \left\lfloor {\frac{{50}}{5}} \right\rfloor + \left\lfloor {\frac{{50}}{{{5^2}}}} \right\rfloor + \underbrace {\left\lfloor {\frac{{50}}{{{5^3}}}} \right\rfloor \, + \ldots }_0 = 10 + 2 = 12\,\,$$ Full explanation: https://www.beatthegmat.com/if-n-is-the-greatest-positive-integer-for-which-5-n-is-a-fac-t304112.html#819465 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 25 Apr 2015 Posted: 2012 messages Followed by: 14 members Upvotes: 43 swerve wrote: What is the largest value of non-negative integer N for which 10^N is a factor of 50!? A. 5 B. 6 C. 12 D. 15 E. 20 The OA is C Source: GMAT Prep To determine the largest value of N, we need to determine how many times 10 divides 50!. Since 10 breaks into primes of 5 and 2, and since there are there are fewer 5s in 50! than 2s, we can find the number of 5s and thus be able to determine the number of 5-and-2 pairs. To determine the number of 5s within 50!, we can use the following shortcut in which we divide 50 by 5, then divide the quotient of 50/5 by 5 and continue this process until we no longer get a nonzero quotient. 50/5 = 10 10/5 = 2 Since 2/5 does not produce a nonzero quotient, we can stop. The final step is to add up our quotients; that sum represents the number of factors of 5 within 50!. Thus, there are 10 + 2 = 12 factors of 5 in 50! This also indicates that there are twelve 5-and-2 pairs, and hence there are 12 factors of 10 within 50! Answer: C _________________ Scott Woodbury-Stewart Founder and CEO • 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 • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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