How about 30 seconds?
This is how I did it. First realize that n(n+1)(n+2) are three consecutive numbers, and they can range from 1 to 96. Using pattern recognition.
Now I am going to write down the first 16 numbers, (I will tell you in a second why this makes it quick to do). I will be calling the product of three consecutive numbers a group.
1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16
Now we have to find the divisibility of groups of 3 consecutive numbers with 8.
1.
2.3.
4.5.
6.7.
8.9.10.11.12.13.14.15.16
Observe that groups starting with even numbers (2,4,6,8 etc) all are divisible by 8 (by groups I mean 2*3*4; 4*5*6 ; 6*7*8; 8*9*10);
1.2.
3.4.
5.6.
7.8.9.10.11.12.13.14.15.16
Also observe that, for all odd numbers (1,3,5,7), the groups starting with them are not divisible by 8 except for the odd number just less than 8. That is 1*2*3, 3*4*5, 5*6*7 are not divisible by 8 but 7*8*9 is (because 8 itself is a factor).
Now observe that this pattern repeats itself again, (that is for groups of 9,11,13 not divisible, groups of 10,12,14,16 and 15 are divisible).
So we find a repeating pattern of 8 where 5 groups are divisible and 3 numbers are not. therefore the probability = 5/8.
Let me know if this helps
