Problem Solving

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.3/8

My workout….
Total no. of outcomes for n(n+1)(n+2) (where n can be any integer from 1 to 96.
Therefore;
1*2*3
2*3*4
3*4*5
4*5*6
5*6*7….
94*95*96
95*96*97
96*97*98
(I hope I am right bcoz ‘n’ can be any integer a the max of 96. )
So there are 96 possible outcomes.

Total no of favourable outcomes.
(All the multiples of 8 till 96) So
6*7*8 14*15*16 94*95*96
7*8*9 15*16*17 95*96*97
8*9*10 16*17*18……….. 96*97*98
So, interger 8 will have 3 outcomes
Interger 16 will have 3 outcomes
Interger 96 will have 3 outcomes
(Which makes 12*3 outcomes = 36)
Plus,
2*3*4 = 24 and divisible by 8
Total no of possible outcomes are 37

And the probability is 37/96

I don’t know where I am going wrong…..Can anyone shed some light on this please…

My answer is 1/2. I did by substituting the valiues

1/2 is not the correct answer. Could someone help me out with this question. THanks

the answer is 1/2 and we do not need so much calculations to come to the answer.

n will either be even or odd.

N even
n * (n+1) * (n+2)
n will wither be multiple of 2 or 4

case 1
n even, multiple of 2
(n+2) will be multiple of 4
so n * (n+2) will be multiple of 8

case 2
n even, multiple of 4
(n+2) will be multiple of 2
so n * (n+2) will be multiple of 8

case 3
n odd
n + 1 will be even
n + 2 will be odd
so n * (n+1) * (n+2) will be multiple of 2 or 4 but not of 8.

Hence for even nos it will be divisible by 8 and for odd it will not be.
so Ans = 1/2 [probabilility of even nos]

Font color: Well, it's a GMAT Prep quesn:

and here's the expln:

for n b/w 1and 96 incl. => any two consecutive even numbers would be multiple pf 8.
thus, if n is even => n+2 is also even thus, div by 8
now, there are 48 such possibilities.

now, if n is odd then n+1 has to be a multiple of 8 and there are 12 multiples of 8 till 96, thus when n is odd, there r 12 nos. div by 8

thus total probability is: ( 48+12 )/ 96 = 5/8

and that's the OA

Answer is 5/8 (had to double check in excel)

Starting with the first 8 numbers (it repeats after that)

1 * 2 * 3 = No
2 * 3 * 4 = Yes
3 * 4 * 5 = No
4 * 5 * 6 = Yes
5 * 6 * 7 = No
6 * 7 * 8 = Yes
7 * 8 * 9 = Yes
8 * 9 * 10 = Yes

5/8

2 Evens = Yes
any combo with an 8 = Yes