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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 ## Prime Number and Divisors ##### This topic has 1 expert reply and 1 member reply ## Prime Number and Divisors If n and p are different positive prime numbers, which of the integers n^4, p^3 , and np has (have) exactly 4 positive divisors? (A) n^4 only (B) p^3 only (C) np only (D) n^4 and np (E) p^3 and np Answer is E ### GMAT/MBA Expert GMAT Instructor Joined 02 Jun 2008 Posted: 2527 messages Followed by: 352 members Upvotes: 1090 GMAT Score: 780 smclean23 wrote: If n and p are different positive prime numbers, which of the integers n^4, p^3 , and np has (have) exactly 4 positive divisors? (A) n^4 only (B) p^3 only (C) np only (D) n^4 and np (E) p^3 and np If n and p are different primes: The divisors of p^3 will be 1, p, p^2 and p^3 (four divisors). The divisors of n^4 will be 1, n, n^2, n^3 and n^4 (five divisors). The divisors of np will be 1, n, p and np (four divisors). You don't need to do all of that if you know how to count divisors from a prime factorization. If you have a prime factorization of x, then you can count the number of divisors of x: -look only at the powers in the prime factorization -add one to each power and multiply. So the number p^3 has 3+1 = 4 divisors, and np has (1+1)(1+1) = 2*2 = 4 divisors. A few more examples- if p and q are different primes, the number (p)*(q^5) would have (1+1)*(5+1) = 2*6 = 12 divisors. The number (p^99)*(q^99) would have (99+1)*(99+1) = 100*100 = 10,000 divisors. _________________ If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com Newbie | Next Rank: 10 Posts Joined 31 Jul 2008 Posted: 5 messages This is the clearest example I have come across in more than two decades Ian Stewart wrote: smclean23 wrote: If n and p are different positive prime numbers, which of the integers n^4, p^3 , and np has (have) exactly 4 positive divisors? (A) n^4 only (B) p^3 only (C) np only (D) n^4 and np (E) p^3 and np If n and p are different primes: The divisors of p^3 will be 1, p, p^2 and p^3 (four divisors). The divisors of n^4 will be 1, n, n^2, n^3 and n^4 (five divisors). The divisors of np will be 1, n, p and np (four divisors). You don't need to do all of that if you know how to count divisors from a prime factorization. If you have a prime factorization of x, then you can count the number of divisors of x: -look only at the powers in the prime factorization -add one to each power and multiply. So the number p^3 has 3+1 = 4 divisors, and np has (1+1)(1+1) = 2*2 = 4 divisors. A few more examples- if p and q are different primes, the number (p)*(q^5) would have (1+1)*(5+1) = 2*6 = 12 divisors. The number (p^99)*(q^99) would have (99+1)*(99+1) = 100*100 = 10,000 divisors. • Award-winning private GMAT tutoring Register now and save up to$200

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