n(n + 1)(n + 2) is a product of 3 consecutive integers.

if n is even, n(n + 1)(n + 2), will be divisible by 8.

even integers from 1-96 inclusive = (96-2)/ 2 = 94/2 = 47, 47+1 = 48

also if n is odd and 1 less than multiple 8, n(n + 1)(n + 2) will be divisible by 8, because this will have at least 1 multiple of 8.

multiples of 8 from 1-96 = (96-8)/8 = 11, 11+1 = 12


total probability = 48+12 = 60/96 = 10/16 = 5/8.

If n is an integer from 1 to 96 (inclusive), what is the probability that n*(n+1)*(n+2) is divisible by 8?

A.1/4
B.1/2
C.5/8
D.7/8
E.3/4

n(n+1)(n+2) = the product of 3 consecutive integers.
WRITE IT OUT and LOOK FOR A PATTERN.

1*2*3
2*3*4
3*4*5
4*5*6
5*6*7
6*7*8
7*8*9
8*9*10

9*10*11
10*11*12
11*12*13
12*13*14
13*14*15
14*15*16
15*16*17
16*17*18

Each of the products in red is a multiple of 8.
The two examples above imply the following:
Of every 8 products, exactly 5 will be a multiple of 8.

Thus, the probability that n(n+1)(n+2) will be a multiple of 8 = 5/8.

The correct answer is C.

Alternate approach:

Case 1: n(n+1)(n+2) = even*odd*even = multiple of 8:
Since every other even integer is a multiple of 4, the product here will always include an even integer and a multiple of 4, resulting in a multiple of 8.
Thus, n can be any even integer between 1 and 96.
96/2 = 48 favorable choices for n.

Case 2: n+1 is a multiple of 8:
The product will be a multiple of 8 if n+1 is 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 that there are 12 more favorable choices for n.

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

For a similar problem, check here:

https://www.beatthegmat.com/probability-t116280.html

Thank you both.

For any even number n, n(n+2) will always be divisible by 8. This is because, either one of n or (n+2) will always be a multiple of 4.

Therefore we just need to find the probability of n being a even number. 48/96.

Also in case n is 7,15 or 23, then n+1 becomes a multiple of 8, so we have to count all numbers that are one ess than multiples of 8, thats 96/8 = 12 numbers

so 48+12 =60
probability is 60/96