Probability

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reachac: Legendary Member; Posts: 572; Joined: Fri Feb 22, 2008 9:25 am; Thanked: 21 times

by reachac » Sat Jul 19, 2008 12:21 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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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

Re: Probability

by parallel_chase » Sat Jul 19, 2008 1:47 am

reachac 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

I think its 1/2. Whats the OA?

n(n+1)(n+2) for this to be divisible by 8 the condition is n should be even

[96-2]/ 2 + 1 = 48

48/96 = 1/2

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Mani_mba: Master | Next Rank: 500 Posts; Posts: 267; Joined: Wed Jul 16, 2008 12:20 amFollowed by: 1 member; Thanked: 4 times

by Mani_mba » Sat Jul 19, 2008 2:20 am

IMO, the answer is B 3/8.

If n(n+1)(n+2) to be divisible by 8, either
a) n should be divisible by 8
b) n+1 should be divisible by 8
c) n+2 should be divisible by 8.

Taking the first case. n can take values
8,16,24,32,40,48,56,64,72,80,88,96( totally 12 )
Taking the second case, n+1 can take values
7,15,.......................................,95 (totally 12)
Taking the third case, n+2 can take values
6,14,.......................................,94 (totally 12)

so totally 36 values out of 96 , n can have to satisfy the condition.
36/96 => 3/8.

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Sat Jul 19, 2008 3:29 am

Mani_mba wrote:IMO, the answer is B 3/8.

If n(n+1)(n+2) to be divisible by 8, either
a) n should be divisible by 8
b) n+1 should be divisible by 8
c) n+2 should be divisible by 8.

Taking the first case. n can take values
8,16,24,32,40,48,56,64,72,80,88,96( totally 12 )
Taking the second case, n+1 can take values
7,15,.......................................,95 (totally 12)
Taking the third case, n+2 can take values
6,14,.......................................,94 (totally 12)

so totally 36 values out of 96 , n can have to satisfy the condition.
36/96 => 3/8.

I think you have exaggerated the problem. You have consider the product of n(n+1)(n+2) not individually. Let me know what you think.

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classic: Newbie | Next Rank: 10 Posts; Posts: 5; Joined: Mon Jun 16, 2008 9:54 pm

by classic » Sat Jul 19, 2008 4:09 am

When n is even it will always be divided by 8

e.g when n=2 then we have 2*3*4

So all together we have 48 even umbers

Now when n= (K*8 -1) where k is natural number it will be divisible by 8

e.g n=7 we will have 7*8*9

so Prob= 60/96=5/8

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Mani_mba: Master | Next Rank: 500 Posts; Posts: 267; Joined: Wed Jul 16, 2008 12:20 amFollowed by: 1 member; Thanked: 4 times

by Mani_mba » Sat Jul 19, 2008 4:37 am

I agree to you Parallel_chase. I missed it.

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reachac: Legendary Member; Posts: 572; Joined: Fri Feb 22, 2008 9:25 am; Thanked: 21 times

by reachac » Sat Jul 19, 2008 4:45 am

I made the same mistake, got 1/2 as the ans.
Correct ans is 5/8.

Thanks @classic...
Thanks ppl

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Sat Jul 19, 2008 12:04 pm

reachac wrote:I made the same mistake, got 1/2 as the ans.
Correct ans is 5/8.

Thanks @classic...
Thanks ppl

The answer is indeed 5/8. That is the most silliest mistake I have ever made.

Thanks for the question.

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evansbd: Senior | Next Rank: 100 Posts; Posts: 52; Joined: Tue Jul 01, 2008 10:50 am

Hold on ....

by evansbd » Tue Jul 22, 2008 12:46 pm

How are we getting 60 total numbers that satisfy this problem?

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

Re: Hold on ....

by parallel_chase » Tue Jul 22, 2008 12:54 pm

evansbd wrote:How are we getting 60 total numbers that satisfy this problem?

Integer which satisfies condition n(n+1)(n+2) is all the even numbers between 1-96

There are total of 48 even integers between 1-96

also you have to consider the possibility what if n is 7 then 7*8*9 will also be divisible by 8

Therefore 8*12 = 96 meaning there are 12 odd integers which satisfies the above condition.

Hence 12+48 = 60

60/96 = 5/8

Hope this helps.

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evansbd: Senior | Next Rank: 100 Posts; Posts: 52; Joined: Tue Jul 01, 2008 10:50 am

I get it but I don't see it....

by evansbd » Wed Jul 23, 2008 5:37 am

I see the solution that you need to find odd numbers that satisfy the criteria. However, on a real GMAT, I easily saw the 48 even numbers but I don't see (intuitively) how ppl are seeing that starting with 7 you can see other odds that will meet the criteria then you end up with 12.

What are the 12 numbers that meet the odd criteria and is there an intuitive way to find them quickly?

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Wed Jul 23, 2008 5:47 am

There is no intuitive way of doing anything on GMAT. Always rely on the details given in the question. Everything has a reason and logic.
It is very necessary to understand the logic and reasoning behind the question.

Now coming to your question.

Odd numbers are
7 (7*8*9)
15(15*16*17)
23(23*24*25)
31(31*32*33)
+
+
+
95(95*96*97)

do you see the method in the above sequence, every multiple of 8 and the number before that.

Hope this helps.

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Oct 24, 2019 12:34 pm

reachac 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

First recognize that n, n+1 and n+2 are 3 CONSECUTIVE INTEGERS.

Now let's make some observations:

When n = 1, we get: (1)(2)(3), which is NOT divisible by 8
n = 2, we get: (2)(3)(4), which is DIVISIBLE BY 8
n = 3, we get: (3)(4)(5), which is NOT divisible by 8
(4)(5)(6), which is DIVISIBLE BY 8
(5)(6)(7), which is NOT divisible by 8
(6)(7)(8), which is DIVISIBLE BY 8
(7)(8)(9), which is DIVISIBLE BY 8
(8)(9)(10), which is DIVISIBLE BY 8
-----------------------------
(9)(10)(11), which is NOT divisible by 8
(10)(11)(12), which is DIVISIBLE BY 8
(11)(12)(13), which is NOT divisible by 8
(12)(13)(14), which is DIVISIBLE BY 8
(13)(14)(15), which is NOT divisible by 8
(14)(15)(16), which is DIVISIBLE BY 8
(15)(16)(17), which is DIVISIBLE BY 8
(16)(17)(18)which is DIVISIBLE BY 8
-----------------------------
.
.
.
The pattern tells us that 5 out of every 8 products is divisible by 8.
So, [spoiler]5/8[/spoiler] of the 96 products will be divisible by 8.
This means that the probability is 5/8 that a given product will be divisible by 8.

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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