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Coprimes below 'n'

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Joined: 11 Aug 2009
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Coprimes below 'n' [#permalink] New post 11 Aug 2009, 11:53
00:00
A
B
C
D
E

Difficulty:

(N/A)

Question Stats:

50% (04:23) correct 50% (01:43) wrong based on 3 sessions
Q) How many numbers exist below 2100 such that the only common factor between that number and 2100 is 1?
a) 480
b) 343
c) 326
d) 540
e) 700

The OA is given as
[Reveal] Spoiler:
A


Please explain how to proceed on this one.
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Joined: 25 Jul 2009
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Schools: NYU, NUS, ISB, DUKE, ROSS, DARDEN
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Re: Coprimes below 'n' [#permalink] New post 11 Aug 2009, 12:12
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I guess you posted this query in the wrong thread! :)

Anyways, I dont think such a question will ever appear on the GMAT. This is a direct application of Euler-Totient Function (Refer http://2000clicks.com/MathHelp/NumberFa ... nction.htm)

To apply this function to the question posted, we will have to factorize 2100:
=> 2100 = 2^2 * 3 * 5^2 * 7
=> 2100 has 4 prime factors viz. 2,3,5 & 7

According to Euler-Totient Fn the number of numbers below 2100 that have 1 as the only common factor with 2100 are:
=> 2100 * (1 - 1/2) * (1 - 1/3) * (1 - 1/5) * (1 - 1/7)
=> 2100 * 1/2 * 2/3 * 4/5 * 6/7
=> 480

ANS: A


PS: Where did you get this question?
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Intern
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Joined: 11 Aug 2009
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Kudos [?]: 1 [0], given: 6

Re: Coprimes below 'n' [#permalink] New post 11 Aug 2009, 12:22
Thats brilliant! Thanks!

How can you bew so sure that such a question will never appear on the GMAT?
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Re: Coprimes below 'n' [#permalink] New post 11 Aug 2009, 12:36
Good explanation!!!
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Joined: 25 Jul 2009
Posts: 117
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Re: Coprimes below 'n' [#permalink] New post 11 Aug 2009, 14:09
sergbov123 wrote:
Thats brilliant! Thanks!

How can you bew so sure that such a question will never appear on the GMAT?



Because there is no other way of solving this sum within 2 minutes and obviously GMAC does not expect the test taker to know about Euler-Totient Fn. I mean this isnt a Math Olympiad man!
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Re: Coprimes below 'n'   [#permalink] 11 Aug 2009, 14:09
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