Probability

Expert replies

dvalenz: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Wed Jul 20, 2011 11:40 am

by dvalenz » Fri Mar 29, 2013 9:37 am

If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n+1)(n+2) will be divisible by 8?

Thanks

Like this post

Source: Beat The GMAT — Problem Solving | Fri Mar 29, 2013 9:37 am

srcc25anu: Master | Next Rank: 500 Posts; Posts: 423; Joined: Fri Jun 11, 2010 7:59 amFollowed by: 2 members; Location: Seattle, WA; Thanked: 86 times

by srcc25anu » Fri Mar 29, 2013 9:47 am

there will be 2 cases in which n * n+1 * n+2 will be divisible by 8
CAse 1: N+1 is divisible by 8. How many factors of 8 are there from 1-96
that will be: 96/8 = 12 nos {8,16,24 .... 88,96}

Case 2:N = even because if n = even, n+2 will also be even
any 2 even numbers when multiplied will contain a 8 between them. eg 2 * 4 will contain 2^3
so essentially we need to know how many enen numbers are there between 1-96 and that will be 48 Nos

Total of such selections of N = 60 Nos
Total integers between 1-96 = 96 nos
hence prob. = 60 / 96 = 5/8
hence D

Like this post

GMAT/MBA Expert

Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Fri Mar 29, 2013 10:34 am

Note that n, (n + 1), and (n + 2) are 3 consecutive integers.
If the product of three consecutive integers are divisible by 8, then the following two scenarios are possible...

1. n is even --> (n + 1) is odd and (n + 2) is even --> Either n or (n + 2) will be divisible by 4
2. (n + 1) is divisible by 8 --> n and (n + 2) are odd

Now, the above two scenarios are mutually exclusive, i.e. no three consecutive integers can be part of both the scenarios. So, there are no repetitions and we can easily calculate number of triplets in each case and add them to get the total number of triplets.

Case 1: n is even
Number of even integers between 1 ans 96, inclusive = {(96 - 2)/2} + 1 = 48

Case 2: (n + 1) is divisible by 8
n + 1 = 8a, where a â‰¥ 1
n = 8a - 1
8a - 1 â‰¤ 96
8a â‰¤ 97
a â‰¤ 12.1 implies 12 integers.

Total number of n = (48 + 12) = 60
Therefore, probability that n(n + 1)(n + 2) will be divisible by 8 = 60/96 = 5/8

The correct answer is D.

Anju Agarwal
Quant Expert, Gurome

Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.

Â§ GMAT with Gurome Â§ Admissions with Gurome Â§ Career Advising with Gurome Â§

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Mar 29, 2013 1:09 pm

dvalenz wrote:If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n+1)(n+2) will be divisible by 8?

Thanks

n(n+1)(n+2) = the product of 3 consecutive integers.

For many test-takers, the most efficient approach will be to WRITE IT OUT and LOOK FOR A PATTERN.

1*2*3
2*3*4
3*4*5
4*5*6
5*6*7
6*7*8
7*8*9
8*9*10

9*10*11
10*11*12
11*12*13
12*13*14
13*14*15
14*15*16
15*16*17
16*17*18

Each of the products in red is a multiple of 8.
The two examples above imply the following:
Of every 8 products, exactly 5 will be a multiple of 8.

Thus, the probability that n(n+1)(n+2) will be multiple of 8 = 5/8.

For a similar problem -- along with an alternate approach -- check my 2 posts here:

https://www.beatthegmat.com/probability-t116280.html

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

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