Algebra

Expert replies

Aman verma: Master | Next Rank: 500 Posts; Posts: 304; Joined: Wed Jan 27, 2010 8:35 amFollowed by: 3 members; Location: International Space Station; Thanked: 11 times

by Aman verma » Thu Aug 11, 2011 5:02 am

Q: The greatest possible divisor of [3^(2n+3)] - 24n -27 for every n belonging to natural number, which necessarily divides is :

a)24

b)64

c)96

d)97

e)none of these

Now the problem has something to do with the formula a^3 + b^3 + c^3 ,maybe where a+b+c = 0, but I am still unable to figure out the solution.

800. Arjun's-Bird-Eye

Like this post

Source: Beat The GMAT — Problem Solving | Thu Aug 11, 2011 5:02 am

Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Thu Aug 11, 2011 5:26 am

Hi,
I don't know about the method you stated.
My solution:
[3^(2n+3)] - 24n -27 = 27.9^n - 24n - 27 = 27(9^n - 1) -24n
9^n - 1 = (8+1)^n - 1 = (8^n + n.8^(n-1) + ... + nC2.8^2 + n.8 + 1) - 1 = 64*k +8n
So, 27(9^n - 1) -24n = 27(64k + 8n) - 24n = 27*64k + 24*9n - 24n = 27*64k + 24*8n
= 24*8(9k+n)
24*8 = 192
Hence, E

P.S. I guess these questions are tougher than those required for GMAT. Please make sure you are working with correct sources.

Cheers!

Things are not what they appear to be... nor are they otherwise

Like this post

gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

by gmatboost » Thu Aug 11, 2011 7:38 am

I would like to second Frankenstein's recommendation that you consider working with other sources, since this is definitely not a GMAT question.

Greg Michnikov, Founder of GMAT Boost

GMAT Boost offers 250+ challenging GMAT Math practice questions, each with a thorough video explanation, and 100+ GMAT Math video tips, each 90 seconds or less.
It's a total of 20+ hours of expert instruction for an introductory price of just $10.
View sample questions and tips without signing up, or sign up now for full access.

Also, check out the most useful GMAT Math blog on the internet here.

Like this post

GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Thu Aug 11, 2011 7:46 am

Aman verma wrote:Q: The greatest possible divisor of [3^(2n+3)] - 24n -27 for every n belonging to natural number, which necessarily divides is :

Well, there is a pretty much faster and easier way to solve this.
The question asks to determine the greatest possible divisor of [3^(2n+3)] - 24n - 27 for every n belonging to natural number. So we can pick the lowest value possible for n, which is 1 and analyze.

For n = 1, [3^(2n+3)] - 24n - 27 = 3^5 - 24 - 27 = 243 - 51 = 192

Thus, 192 is the greatest possible divisor.

The correct answer is E.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/

Like this post

Aman verma: Master | Next Rank: 500 Posts; Posts: 304; Joined: Wed Jan 27, 2010 8:35 amFollowed by: 3 members; Location: International Space Station; Thanked: 11 times

by Aman verma » Thu Aug 11, 2011 9:14 am

Hi!! all, thanks for the response. Now I can figure out my mistake. I was amazed by the Frankenstein's approach particularly the Binomial Expansion used, I was expecting a simpler approach. Nevertheless, this is one of the several ways to approach a problem . One should have all the tools in his arsenal for better preparation. Also I was inspired by Anurag sir's approach. So simple and effective.

Now regarding Gmatboost's concerns, I agree that this might not replicate exactly a GMAT question, but nevertheless one should prepare for any eventuality. The GMAT may throw an awkward question when one takes the actual test. I would like to mention my own experience- when I took the test one question from CONIC SECTION involving Focus, Latus Rectum did appeared and I was completely outwitted. Although Conic Section is not in the syllabus for GMAT, it did appeared on the actual test. It might have been an experimental question but nobody can tell beforehand which ones are experimental. So I would say one should prepare for any sort of eventuality. Besides an extra effort never hurts anyone !!

800. Arjun's-Bird-Eye

Like this post

gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

by gmatboost » Thu Aug 11, 2011 10:35 am

I feel pretty strongly that it's not worth spending time worrying about topics that are not in the Official Guide's math review chapter (e.g. binomial distribution, conic sections). Of course there is nothing wrong with knowing them, but I maintain that you will not see a question on which you MUST know them.

Also, I would caution that plugging in n = 1 to get 192 does NOT in itself prove that 192 "necessarily divides" 3^(2n+3) - 24n - 27

Like this post

Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Thu Aug 11, 2011 10:43 am

gmatboost wrote: Also, I would caution that plugging in n = 1 to get 192 does NOT in itself prove that 192 "necessarily divides" 3^(2n+3) - 24n - 27

Exactly! This is not justified and cannot be generalized although it works in this particular example.

Cheers!

Things are not what they appear to be... nor are they otherwise

Like this post

GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Thu Aug 11, 2011 11:19 am

Hi Greg and Frankenstein!

Number theory as a topic is very much logical and manipulative one. Fluke or luck has a very small role to play here. In my experience I've seen these problems (particularly those which involves integers or natural numbers) do follow a pattern which is true for all the integers in the range if they are true for the lowest permitted ones. And almost always there exist a mathematical logic behind them.

In this case also my argument is purely based on mathematical logic which does not involve any concept beyond GMAT (like binomial theorem etc). My only 'mistake' was that I didn't show all the logical details to support my claim. Anyway here it goes...

We can rewrite the expression as follows,

T(n) = 3^(2n+3) - 24n - 27
= 9*[3^{2(n - 1) + 3} - 24(n - 1) - 27] + 192n
= 9*T(n - 1) + 192n

So for n = 1, the expression, i.e. T(1) = 192
Therefore,

for n = 2, T(2) = 9*T(1) + 192*2 = 9*192 + 192*2 = 11*192
for n = 3, T(3) = 9*T(2) + 192*3 = 9*11*192 + 192*3 = 102*192
... so on and so forth

So in this case plugging in n = 1 to get 192 does in itself prove that 192 "necessarily divides" 3^(2n+3) - 24n - 27 for all possible values of n.

Like this post

Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Thu Aug 11, 2011 11:49 am

Hi Anurag,
Thanks for posting. There is no disputing your knowledge/experience(in fact any experts). But, without proof, which has been posted now, at least some of the readers won't be convinced as we can always find exceptions or counter examples for similar cases. All of us definitely want to know the underlying concept and make sure it works *always*. Thanks again!

Cheers!

Things are not what they appear to be... nor are they otherwise

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review