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If an integer n is to be chosen at random from integers 1 to

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Manager
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If an integer n is to be chosen at random from integers 1 to [#permalink]  05 Dec 2004, 15:08
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3. If an integer n is to be chosen at random from integers 1 to 96 inclusive, what is the probability that n(n+1)(n-1) will be divisible by 8?

a. 1/4
b. 3/8
c. 1/2
d. 5/8
e. 3/4

Pls explain
Manager
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B. 3/8

n(n+1)(n-1) is divisible by 8 if any or all of its terms are divisible by 8.
Lets assume X=[1,96];
The probability P that X is divisible by 8 is P = 1/8.
X could be equal to n or (n-1) or (n+1), so there are 3 possible favorable outcomes out of 8 possible outcomes.
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For some unknown reason 3/8 is not the answer!
Director
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5/8.

Any odd number value of n would be divisible by 8. ex. 3(3+1)(3-1) or 5(5+1)(5-1) etc. - we have 48 odd numbers between 1 and 96.

Also numbers such as 8, 16, 24....are divisible by 8. There are 96/8 = 12 such numbers.

In total 48 + 12 = 60 numbers.

prob = 60/96 = 5/8.
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Anybody else wanna try?
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I'd go with C 1/2
We need the expression n(n+1)(n-1) to contain 2^3 to be divisible by 8
Plug in 1, 2, 3, 4... and you will see that every other 2nd number starting with 1 will have 2^3 in it.
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Same answer as venksune, all odd numbers + multiple of 8 = 48+12=60 out of 96
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quest: n(n+1)(n-1) - to check n (from 1~96) divisible by 8.

I had re-arranged the equation and saw that it is a consecutive number style.

(n-1) n (n+1).

Multiples of 4 is also a multiple of 8 (if the sequence does not start with an odd number).

Example :
2 x 3 x 4
3 x 4 x 5
4 x 5 x 6

Hence 96/4=24
24 x 2 =48 - 2 multiples 4 are divisible of 8 and the other one is not.

Therefore Prob = 48/96 =1/2 (Ans C)
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Good question. I just read Venksune's approach and I have to agree
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Paul

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the total cases : 96
favorable :
for all multiples of 8 we have 3 triplets i.e in total 12 * 3 = 36 triplets
for all odd multiples of 4 we have 2 triplets so 12 *2 = 24

OA is E.
the key here is to determine the favorable outcomes for the task: either 12x3, either 12x2, either 12 events
(n-1)n(n+1)
12 12 12
12 12
12
The total sum of favorable outcomes is 12x3 + 12x2 + 12 = 72
So it is 72/96 = 3/4.
Manager
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Thanks guys, but the OA is D. Saw this question on another site and thot it was interesting!
Director
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venksune wrote:
5/8.

Any odd number value of n would be divisible by 8. ex. 3(3+1)(3-1) or 5(5+1)(5-1) etc. - we have 48 odd numbers between 1 and 96.

Also numbers such as 8, 16, 24....are divisible by 8. There are 96/8 = 12 such numbers.

In total 48 + 12 = 60 numbers.

prob = 60/96 = 5/8.

Venksune, I took the same approach as you but I dont get 5/8. This is the problem I have can you help?

In your approach you seem to be counting the odd number 1. But the set (1,2,0) does not exist because the range is from 1 to 96. Besides 2 is not not is not divisible by 8. So, I get 47 + 12 = 59.

What am I missing??
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we are choosing a number n from 1 to 96. If we choose n=1, then we have (n-1)(n)(n+1)=0. However, 0/8 is still 0. 0 is divisible by 8. then next odd number is 3 in which case (n-1)(n)(n+1) would be 2*3*4 = 24, which is divisble by 8. Same is the case with 5...so on - totalling to 48 of them that are divisible by 8.

I didn't quite get your reference to 2.

I feel that the question will have NO ambiguity if it reads n(n+1)(n+2)
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gayathri, the result set is (1,2,0) meaning that when you take 1, you get 1*(1+1)*(1-1) = 1*2*0 = 0
0 IS a multiple of 8
Hence, take every odd number from the domain, 1 to 96, you will have a number divisible by 8 and there are 48 of those. But as Venksune explained, every even number that is a multiple of 8 will also give a result divisible by 8 and there are 12 of those.

48+12 = 60/96 = 5/8
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Paul

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Thanks for the explanation Paul & Venksune. I must have had a brain freeze.

By some stroke of mathematical genius I came uo with 1*2*0 =2 !!!
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