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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 smallest positive integer tagged by: VJesus12 ##### This topic has 1 expert reply and 0 member replies ## What is the smallest positive integer What is the smallest positive integer n for which $$\sqrt{n\cdot7!}\ is\ an\ integer?$$ A. 14 B. 35 C. 70 D. 105 E. 210 The OA is the option B. How can I find the value of n? Should I try number by number? Experts, may you give me some help, please? ### GMAT/MBA Expert GMAT Instructor Joined 04 Oct 2017 Posted: 551 messages Followed by: 11 members Upvotes: 180 Top Reply Hello Vjesus12. Let's see your question. You could try number by number or solve it as follows: $$\sqrt{n\cdot7!}=\sqrt{n\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}=2\sqrt{n\cdot7\cdot(3\cdot2)\cdot5\cdot3\cdot2}$$ $$2\sqrt{n\cdot7\cdot2\cdot3\cdot5\cdot3\cdot2}=2\sqrt{n\cdot7\cdot4\cdot9\cdot5}=2\cdot2\cdot3\sqrt{n\cdot7\cdot5}=12\sqrt{n\cdot7\cdot5}.$$ Now, this last number is an interger when the prime factorizazion has an odd number of 5's and a 7's. On the given list, the smallest number that holds this condition is 35. So, the correct answer is B=35. I hope this explanation may help you. I'm available if you'd like a follow-up. Regards. _________________ GMAT Prep From The Economist We offer 70+ point score improvement money back guarantee. Our average student improves 98 points. Free 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now. • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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