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Probability

by deepakb » Tue Oct 05, 2010 7:27 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?
A. 1/4

B. 3/8

C. 1/2

D. 5/8

E. 3/4

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by neerajkumar1_1 » Tue Oct 05, 2010 7:51 am
IMO: D

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by shovan85 » Tue Oct 05, 2010 7:52 am
IMO D

This question can be taken as how many even numbers + how many multiples of 8 can be chosen
= 48+12
=60

Total numbers 96

So probability 60/96

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by GMATGuruNY » Tue Oct 05, 2010 7:54 am
deepakb 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?
A. 1/4

B. 3/8

C. 1/2

D. 5/8

E. 3/4
Even*odd*even = multiple of 8:
Given 3 consecutive integers {even, odd, even}, the product will always be a multiple of 8.
Thus, n can be any even integer between 1 and 96.
96/2 = 48 favorable choices for n.

n+1 is a multiple of 8:
The product will be a multiple of 8 if n+1 is a multiple of 8 (making an odd integer that is 1 less than a multiple of 8).
Number of multiples of 8 between 1 and 96 = 96/8 = 12.
Thus, there are 12 favorable choices for n+1, implying 12 more favorable choices for n.

Total favorable choices for n = 48+12 = 60.
Favorable choices/Total choices = 60/96 = 5/8.

The correct answer is D.
Last edited by GMATGuruNY on Sun Jul 31, 2011 4:26 am, edited 1 time in total.
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by neerajkumar1_1 » Tue Oct 05, 2010 8:00 am
First...

n(n+1)(n+2) represent 3 consec numbers... and we can divide all the 3 consec number series in even ones and odd ones...

if n is even, then (n+2) will be even, and the product will be divisible by 8
therefore half the series will be multiple of 8... i.e 96/2 = 48

also if n is odd, then (n+2) will be odd, so (n+1) all by itself will have to be a multiple of 8
therefore 96/8 = 12 multiples of 8, which will be exactly the middle number...

hence we have 48 + 12 = 60 series which will be a multiple of 8 out of a total of 96

=60/96 = 5/8