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Register now and save up to $200 Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder 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 • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • Reach higher with Artificial Intelligence. Guaranteed Now free for 30 days Available with Beat the GMAT members only code ## n divisible by 6 tagged by: Brent@GMATPrepNow This topic has 1 expert reply and 3 member replies j_shreyans Legendary Member Joined 07 Aug 2014 Posted: 510 messages Followed by: 5 members Upvotes: 3 #### n divisible by 6 Wed Sep 10, 2014 11:06 pm Is the positive integer n divisible by 6? (1) n^2/180 is an integer. (2) 144/n^2 is an integer OAA ### Top Member GMATinsight Legendary Member Joined 10 May 2014 Posted: 1044 messages Followed by: 23 members Upvotes: 205 Mon Oct 12, 2015 7:46 pm [quote="prada"] GMATinsight wrote: j_shreyans wrote: Statement 2) 144/n^2 is an integer i.e. 2^4 x 3^3 is divisible by n^2 Answer: Option A Sorry for bumping an old thread (linked from GMATPREP) but shouldn't it be 2^4 x 3^2? I think you did the PF of 144? Yes, It was PF of 144. I made a typo mistake. Thanks for taking note of it. Have made the necessary changes. _________________ Bhoopendra Singh & Sushma Jha - Founder "GMATinsight" Testimonials e-mail: info@GMATinsight.com I Mobile: +91-9999687183 / +91-9891333772 To register for One-on-One FREE ONLINE DEMO Class Call/e-mail One-On-One Private tutoring fee - US$40 per hour & for FULL COURSE (38 LIVE Sessions)-US$1000 prada Senior | Next Rank: 100 Posts Joined 08 Dec 2010 Posted: 64 messages Upvotes: 1 Mon Oct 12, 2015 5:12 pm [quote="GMATinsight"] j_shreyans wrote: Statement 2) 144/n^2 is an integer i.e. 2^4 x 3^3 is divisible by n^2 Answer: Option A Sorry for bumping an old thread (linked from GMATPREP) but shouldn't it be 2^4 x 3^2? I think you did the PF of 144? ### Top Member GMATinsight Legendary Member Joined 10 May 2014 Posted: 1044 messages Followed by: 23 members Upvotes: 205 Thu Sep 11, 2014 4:12 am j_shreyans wrote: Is the positive integer n divisible by 6? (1) n^2/180 is an integer. (2) 144/n^2 is an integer OAA Question : Is the positive integer n divisible by 6? Question : Is n Multiple of 2 as well as 3? Statement 1) n^2/180 is an integer. i.e. n^2 = 2^2 x 3^2 x 5 x A where A is any Integer i.e. n is a multiple of 2 as well as 3 SUFFICIENT Statement 2) 144/n^2 is an integer i.e. 2^4 x 3^2 is divisible by n^2 now it is possible for n = 6 (a multiple of 6 also it is possible for n = 4 which is not a multiple of 6 Inconsistent answer therefore NOT SUFFICIENT Answer: Option A _________________ Bhoopendra Singh & Sushma Jha - Founder "GMATinsight" Testimonials e-mail: info@GMATinsight.com I Mobile: +91-9999687183 / +91-9891333772 To register for One-on-One FREE ONLINE DEMO Class Call/e-mail One-On-One Private tutoring fee - US$40 per hour & for FULL COURSE (38 LIVE Sessions)-US\$1000

Last edited by GMATinsight on Mon Oct 12, 2015 7:44 pm; edited 1 time in total

### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
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Thu Sep 11, 2014 6:35 am
j_shreyans wrote:
Is the positive integer n divisible by 6?

(1) nÂ²/180 is an integer.

(2) 144/nÂ² is an integer

OAA
NOTE: This question is VERY similar to this one: http://www.beatthegmat.com/ds-problem-t279217.html

Okay, a lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)

For more on this concept, see the 2nd half of this free video: http://www.gmatprepnow.com/module/gmat-integer-properties/video/825

------------------------------

Okay, onto the question!

Target question: Is the positive integer n divisible by 6?

Statement 1: nÂ²/180 is an integer
This tells us that nÂ² is DIVISIBLE by 180
This means that 180 is "hiding in the prime factorization of nÂ²
180 = (2)(2)(3)(3)(5)
So, nÂ² = (2)(2)(3)(3)(5)(?)(?)(?)(?)

Aside: the (?)'s represent other possible primes in the prime factorization of nÂ²

Rewrite as (n)(n) = [(2)(3)(5)(?)(?)][(2)(3)(5)(?)(?)]
This tells us that we can be certain that n = (2)(3)(5)(?)(?)
At this point it is clear that n is divisible by 6
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 144/nÂ² is an integer
There are several values of n that satisfy this condition. Here are two:
Case a: n = 1. Notice that 144/1Â² = 144, and 144 is an integer. In this case n is NOT divisible by 6
Case b: n = 6. Notice that 144/6Â² = 4, and 4 is an integer. In this case n IS divisible by 6
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

--------------------

Here are a few more questions that require us to understand the relationship between divisibility and prime factorization:
http://www.beatthegmat.com/factors-t272765.html
http://www.beatthegmat.com/do-t-and-12-have-common-factors-t274544.html
http://www.beatthegmat.com/is-xy-a-multiple-of-105-t274522.html
http://www.beatthegmat.com/confused-need-help-here-t271655.html
http://www.beatthegmat.com/multiple-of-990-t272719.html
http://www.beatthegmat.com/if-n-t-3-for-some-positive-integer-t148420.html

Cheers,
Brent

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Brent Hanneson â€“ Founder of GMATPrepNow.com
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