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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: Is positive integer p even? tagged by: Brent@GMATPrepNow ##### This topic has 1 expert reply and 1 member reply ### GMAT/MBA Expert ## Challenge question: Is positive integer p even? ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult Is positive integer p even? (1) 4p has twice as many positive divisors as p has (2) 8p has 3 positive divisors more than p has Answer: A Source: www.gmatprepnow.com Difficulty level: 700+ _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these 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 08 Dec 2008 Posted: 12982 messages Followed by: 1249 members Upvotes: 5254 GMAT Score: 770 Brent@GMATPrepNow wrote: Is positive integer p even? (1) 4p has twice as many positive divisors as p has (2) 8p has 3 positive divisors more than p has Answer: A Source: www.gmatprepnow.com Difficulty level: 700+ ------ASIDE--------------------- Here's a useful rule: If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors. Example: 14000 = (2^4)(5^3)(7^1) So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40 ----------------------------------- Target question: Is positive integer p even? Statement 1: 4p has twice as many positive divisors as p has Since p is a positive INTEGER, we know that p is either EVEN or ODD I'll show that p cannot be odd, which will allow us to conclude that p must be even. If p is ODD, then the prime factorization of p will consist of ODD primes only. We can write: p = (some odd prime^a)(some odd prime^b)(some odd prime^c).... So, the number of positive divisors of p = (a+1)(b+1)(c+1)... Let's let k = (a+1)(b+1)(c+1)... That is, k = the number of positive divisors of p Now let's examine the prime factorization of 4p 4p = (2^2)(a+1)(b+1)(c+1)... So, the number of positive divisors of 4p = (2+1)(a+1)(b+1)(c+1)... = (3)(a+1)(b+1)(c+1)... = (3)(k) So, p has k divisors, and 4p has 3k divisors. In other words, 4p has THREE TIMES as many divisors as p. HOWEVER, we need 4p to have TWICE as many divisors as p. So, we can conclude that p CANNOT be odd, which means p must be even Aside: For example, if p = 2, then it has 2 divisors, and 4p = 8, which has 4 divisors. So, 4p has TWICE as many divisors as p Statement 2: 8p has 3 positive divisors more than p has There are several values of p that satisfy statement 2. Here are two: Case a: p = 1, which means 8p = 8. 1 has 1 divisor (1), whereas 8 has 4 divisors (1, 2, 4, 8). So, 8p has 3 positive divisors more than p has. In this case, the answer to the target question is NO, p is NOT even Case b: p = 2, which means 8p = 16. 2 has 2 divisors (1, 2), whereas 16 has 5 divisors (1, 2, 4, 8, 16). So, 8p has 3 positive divisors more than p has. In this case, the answer to the target question is YES, p is even 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 my video course along with Sign up for free Question of the Day emails And check out all of these 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! ### Top Member Legendary Member Joined 02 Mar 2018 Posted: 1104 messages Followed by: 2 members 4p has twice as many positive divisors as p If p = 6 , then 4p = 4 * 6 = 24 6 has 4 factors and 24 has $$2^3\cdot3^1$$ 4 * 2 = 8 factors If p = 2 then p has 2 divisors 4p = 4*2 = 8 which has four divisors Therefore ; 4p has twice as many divisors as p. Hence, integer p is even , Statement 1 is SUFFICIENT. Statement 2 8p has 3 positive divisors more than p has If p = 1 ; 8p = 8 this means p has one divisor and 8p has four divisor, so 8p has 3 more divisor than p has which means p is NOT even. If p = 2 ; 8p = 16 this means p has two divisors and 8p has 5 divisors although 8p has 5 more divisors more than p has, p is NOW even. Information given is not enough to arrive at a specific answer, hence statement 2 is INSUFFICIENT. Statement 1 alone is SUFFICIENT. $$answer\ is\ Option\ A$$ • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep 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 • Award-winning private GMAT tutoring Register now and save up to$200

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