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 Author: questioner [ Thu Aug 19, 2010 8:59 am ] Post subject: GMAT Number Theory Let a be a positive integer. If n is divisible by 2ª and n is also divisible by 3²ª, then it is possible that n is NOT divisible byA. 6B. 3 × 2ªC. 2 × 3²ªD. 6ªE. 6²ª(E) If n is divisible by 2ᵃ and 3²ᵃ then it must be divisible by least common multiple of 2ᵃ and 3²ᵃ which equals 2ᵃ × 3²ᵃ.Therefore the smallest possible value number n can take is 2ᵃ × 3²ᵃ which is less than answer choice (E), 6²ᵃ = 2²ᵃ × 3²ᵃ. A larger number can not be a divisor of a smaller one so 2ᵃ × 3²ᵃ is NOT divisible by 6²ᵃ. It means, that 6²ᵃ is not necessarily a divisor of n.The right answer is choice (E).---------Why do we conclude from "that this number is smaller than choice (E)," that choice (E) is the right one?

 Author: Gennadiy [ Thu Aug 19, 2010 9:32 am ] Post subject: Re: math: exponents, number theory, factorization We make a conclusion that choice (E) is the right one because we've shown that if n = 2ª × 3²ª (it fully satisfies the question statement) then it is not divisible by 6²ª simply because 6²ª is greater.And furthermore I'd like to go over this question once again in details.The question statement tells us that n is divisible by 2ª and 3²ª, where a is some positive integer. Therefore we can represent n as: n = 2ª × 3²ª × d, where d is some positive integer.Note, that we consider n to be a positive integer for simplicity since divisibility of n or -n is the same.We write the above mentioned formula for n because 2ª and 3²ª are relatively prime. If they where, for example, 10 and 15 (instead of 2ª and 3²ª) then n would equal 2 × 3 × 5 × d because 2 × 3 × 5 is the least common multiple of 10 and 15 while the least common multiple of 2ª and 3²ª is 2ª × 3²ª.Let's return to our formula and consider all the answer choices:n = 2ª × 3²ª × dA. Is n divisible by 6?Yes, because it is clearly divisible by 2 and 3.B. Is n divisible by 3 × 2ªYes, because it is clearly divisible by 3 and 2ª is a multiplier in formula for n.C. Is n divisible by 2 × 3²ªYes, because it is clearly divisible by 2 and 3²ª is a multiplier in formula for n.D. Is n divisible by 6ªYes, because we can rewrite the formula for n:n = 2ª × 3ª × 3ª × d and 6ª = (2 × 3)ª = 2ª × 3ªSo 2ª and 3ª are the multipliers in formula for n.E. Is n divisible by 6²ªIt can be or it can be not. Let us divide n by 6²ª:n / 6²ª = (2ª × 3ª × 3ª × d) / (2ª × 2ª × 3ª × 3ª) = d / 3ªSo n / 6²ª = d / 3ªWe clearly see that if d is divisible by 3ª then n is divisible by 6²ª. But if , let's say, d = 1 then n is not.Therefore n MUST BE divisible by the choices (A) – (D) and NOT NECESSARILY by choice (E).

 Author: questioner [ Mon Sep 06, 2010 11:53 am ] Post subject: Re: math (test 3, question 8): exponents, number theory I'm not seeing this:- 6 to the 2a is different than 6 to the a how?- both 6's are multiples of each of the primes provided and how is C not the same? (2 × 3 to the 2a?)

 Author: Gennadiy [ Mon Sep 06, 2010 12:09 pm ] Post subject: Re: math (test 3, question 8): exponents, number theory Choice (C), 2 × 3²ª, is much smaller than Choice (E), 6²ª = (2 × 3)²ª = 2²ª × 3²ª.Note that notations (2 × 3)²ª and 2 × 3²ª mean two different numbers.Choices (D) and (E) also differ.6²ª = (6²)ª = 6ª × 6ª or 36ª. It is clearly greater than choice (D), 6ª.

 Author: questioner [ Sat Feb 25, 2012 12:37 pm ] Post subject: Re: math (test 3, question 8): exponents, number theory I did not understand the explanation to this question quite well. Why is it not B for example?

 Author: Gennadiy [ Sat Feb 25, 2012 12:45 pm ] Post subject: Re: math (test 3, question 8): exponents, number theory questioner wrote:Why is it not B for example?Because, if n is divisible by 2ª and n is also divisible by 3²ª, then n MUST be divisible by 3 × 2ª (B). While the question asks us to choose a value that might NOT be a divisor of n.In other words, we don't know what the real value of n is. It can be any value that satisfies "n is divisible by 2ª and n is also divisible by 3²ª". Choices A, B, C, D are divisors of any such value of n. While there is at least one possible value of n (e.g. 2ª × 3²ª) for which E, 6²ª, is NOT a divisor.

 Author: questioner [ Fri Jul 13, 2012 9:33 am ] Post subject: Re: t.1, qt.18: exponents, number theory Not sure I am following the logic here due to the fact that Answer Choice E seems to be larger than Answer Choice D? I could be completely missing the point here, and most likely the case, but if you have some additional explanation that would be really helpful. Would picking numbers at all make sense here? Also, is this a 700-800 level question or more in the range of 600-700? Please let me know when you have an opportunity, thank you very much.