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
what is the probability that n(n + 1)(n + 2) will be divisible by 8?
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Ans B 3/8
Ans. 3/8
5/8. Denominator= 96c1=96
For n(n+1)(n+@) to be even n has to be either even ie 48c1 48 ways..or nis odd and n+2 is divisible by 8 12 ways.. so numerator=48+12=60..prob=60/96=5/8
60/96= 5/8.
Denominator= 96c1=96
FOR EVEN NO. IT IS DIVISIBLE BY 8 I.E. 48c1 = 48
N = Odd and ( N+1) is divisible by 8 that will be 12 nos.
= 60 / 96 = 5 / 8
i am totally clueless about this one. will someone please care to explain?
i meant a little more elaborately. i know the ans now that some of u have explained. but how do u know that this is method to follow???
A
The answer is D: 5/8
If you multiply any even number, n, in the sequence of n(n + 1)(n + 2), the number will be divisible by 8. 96/2 = 48 even numbers
If you multiply any odd number, n, that has a number divisible by 8 in the sequence of n(n + 1)(n + 2), the number will be divisible by eight. 96/8 = 12 for the numbers that will be divisible by eight.
48 + 12 = 60
60 /96 reduced is 5/8
5/8
D) 5/8
B
for n*n+1*n+2 to be divisible by 8 we can have either n as a multiple of 8 or we can have n+1 as a multiple of 8 or we can have n+2 as a multiple of 8.
Now for n we can have 12 values from 1-96, similarly for the other two also we would have 12 values each. Thus the probability would be 36/96 ie 3/8
B
D) 5/8 (All even n’s >= 2 will result in n(n+1)(n+2) being divisible by 8. And all (n+1)’s that are multiple by 8.)
5/8
Take Gmat Team plz giv correct ans and proper solution
Hi Copal,
Official answer is D
D
explanation by sodielectric is perfect
hey, the ans is D.
Answer is 3/4 I think. (E)
A ….i guess…. whenever n is a muliple of 4, n*n+1*n=2 is divisible by 8
Sorry, the correct answer is 1/2 .. whenever n is even, the condition is true
hehe f**k, its a good question, whenever n is even or n = 8t -1 , the condiion is true, so 1/2 for even numbers + 1/8 for 8t-1 numbers i.e.
1/2+1/8 = 5/8
its D
8=2*4
SO NO MUST BE DIV BY 2 N 4 n 8
there r 48 no tht will be divisble by 2
n 24 nos tht will be div by 4
n there r 12 nos tht will be divisible bye 8 n tht will be common nos so we will substract those frm those which r divisible by 2 n 4
so its 48+24-12 =60so the probability is 60/96=5/8
Ans D 5/8
Because one is short on time, I got the right answer by solving this after figuring the probability for 1 to 8 would be the same as the probability for 1 to 96 (96 = 8 * 12). If you list the number for n from 1 to 8, you get 5/8.
i.e.
n=1 : 1x2x3 => No
n=2 : 2x3x4 => Yes
n=3 : 3x4x5 => No
n=4 : 4x5x6 => Yes
n=5 : No
n=6 : Yes
n=7 : Yes
n=8 : Yes
Total Yes: 5/8
The answer is B – 3/8
The earlier response that if n is even then
n (n + ) (n+ 2) is divisible by 96/2 = 48 numbers is not correct.
There are 48 even numbers from 1 to 96.
If n = 10
10 ( 10 + 1 )(10 + 2) = 10.11.12 is not divisible by 8.
There are many such even numbers like this that is no divisible by 8.
Correct Answer acc to me:
for n.(n + 1).(n + 2) to be divisible by 8
either n should be divisble by 8
or n + 1 should be divisible by 8
or n + 2 should be divisible by 8
for n to be divisible by 8 ,it should be a direct multiple of 8
like 8,16,24 … In total 12 numbers
For n + 1 to be divisible by 8, n should be an odd number
Here again there are 12 numbers divisible by 8
7 + 1 = 8
15 + 1 = 8 * 2
23 + 1 = 8 * 3
so on…
For n + 2 similarly there are 12 numbers
6 + 2 = 8
14 + 2 = 16
22 + 2 = 24
so on ..
Adding all the 3 total ( 12 + 12 + 12 )
so 12 * 3 = 36
Probability = Possible outcomes / Total sample
= 36 / 96 = 3/8
I agree with Chand on this analysis.. It shud be 3/8.
Actually…
10 X 11 X 12 is divisible by 8….
1320 / 8 = 165
answer is 3/8. how??
ok let me give my fanda..
of n consecutive natural numbers, 1 and only 1 number is divisible by 8. i mean if we take any 8 consecutive numbers say 1 to 8. only one number(8) is divisible by 8. or u can take any other consecutive numbers say 20 to 28(only 24 divisible by 8).
how i used my fanda?
if in a group of 8 consecutive numbes only 1 number is divisible by 8. so also in a group wid 3 numbrs only 1 number at most can be divisible by 8.
now if we divide 1 to 96 into groups of 8 numbers each. we have 12. now each of the number either ‘n’, ‘n+1′ or ‘n+2′ falls in any one of the 12 groups.
probablity that n is divisible by 8 = 1/8 (n can fall in any group)
probablity that n+1 is divisible by 8 = 1/8 (n+1 can fall in any group)
probablity that n+2 is divisible by 8 = 1/8 (n+2 can fall in any group)
as per our question and as per my fanda(stated above) only 1 number is divisible by 8 ie either n is divisible by 8 OR n+1 is divisible by 8 or n+2 is divisible by 8. and since its an OR case, the required answer will be the some of the three probablities stated above ie
1/8 + 1/8 + 1/8 = 3/8
any criticism about my fanda n method is WELCOME.
or if still unclear, u can mail me at
I agree with Eshu..
Great work.
Dear Eshu,
You forgetting something very important :
N * (N+2) can result in a number divisible by 8 if N is even.
You have to test the divisibility of the product. Not N itself.
Simple example: 2 * 3 * 4 = 24 which is divisible by 8.
Now go back to your notepad and count again.
- RD
D
The answer & explanation provided by
“Sky Surfer
#26. December 3rd, 2007, at 9:46 AM. ”
is gud and apt, as long as the top end number i.e. 96 in this case, remains divisible by 8
all the even numbers would satisfy the equation : 48 instances
n=7, satisfies the equation
n=15, satifies the equation
n=23, satifies the equation…
hence, every 8th odd number from 7 would satisfy the equation:
and there are 12 such cases (96/8 = 12)
Total no. of favourable cases: 48+12
Total cases : 96
probability : 60/96 : 5/8
hm!
If you list the number for n from 1 to 8, you get 5/8.
n=1 : 1×2×3 => No
n=2 : 2×3×4 => Yes
n=3 : 3×4×5 => No
n=4 : 4×5×6 => Yes
n=5 : No
n=6 : Yes
n=7 : Yes
n=8 : Yes
D
Total Yes: 5/8
D is the answer… 5/8
Option D
Answer is B: 3/8
For n, n+1 & n+2 there are 12 nos., divisble by 8 between 1 to 96.
Hence probability will be (12+12+12)/96 = 3/8
Answer appears to be 5/8 .