How did you know that we had to check every 8 numbers exactly?

If we check every 12 numbers (because 12 * 8 = 96 ) then every 12 numbers have 7 valid values.

thus 7 * 8 = 56 which would give incorrect answer.

The key here is that we're looking for products that are divisible by 8.
We start with n = 1 (which is 1 more than a multiple of 8)
Then n = 2 (which is 2 more than a multiple of 8)
.
.
.
Then n = 7 (which is 7 more than a multiple of 8)
Then n = 8 (which is 0 more than a multiple of 8)
Then n = 9 (which is 1 more than a multiple of 8) at this point we can see that the pattern begins repeating itself
.
.
.
etc

Does that help?

Here is how I approached the problem:

n(n+1)(n+2) divisible by 8 means n is divisible by 4 or n+1 is divisible by 8 or n+2 is divisible by 4. Worth noting that whenever n or n+2 is divisible by 4, the other is also an even number, so n(n+2) will be divisible by 8. Also worth noting that every consecutive 3 numbers have at most one multiple of 4 and, of course, alternate odd and even numbers.

Case 1: n divisible by 4: 96/4 = 24 numbers
Case 2: n+2 divisible by 4; similar to case 1: 24 numbers
Case 3: n+1 divisible by 8: 96/8 = 12 numbers

Total: 24+24+12=60

Probability: 60/96=5/8

we see a pattern that when the consecutive no's start with even, we have multiples of 8 but not when n is odd.
But we also have a case where (n+1) is a multiple of 8 i.e. 8,16,24,32 upto 96
So, in this case, the 'n' can be odd but be divisible by 8.
• Ex: 7,8,9
○ 15,16,17
• But this case is already included where (n+2) is multiple of 8
○ 14,15,16: here n is even and included by 1st case.

• Total number of even numbers between 1 to 96 = 48
• Total number of multiples of 9 between 1 to 96 = 12
○ Total = 60
• Probability = 60/96 = 5/8

I have a better solution. carcass's solution is really bad. notice if you pick any EVEN number from 96 and put it into equation. product of n n+1 and n+2 (EVEN-ODD-EVEN) will always be divisible by 8 and will always include more than three 2's. So for N as any even number between 1-96 (there are 48 of them) it is divisible by 8. ( here We get 48 EVEN numbers from for N)


If we take into second condition into consideration which is n is an ODD number. so our equation will be ( ODD-EVEN-ODD). So We have only 1 EVEN to divide 8. Odd's cant not be divided by 8. We can only use 8-16-24-32.....96 . Because for example 14 this time will not be divided by 8. We only need multiples of 8.( We determine here EVEN number not the N) . But we here in this case choose N to be odd. so we can choose n as 7-15-31....95 SO THAT N+1 will be 8-16-24....96. So in this scenario we have 12 ODD numbers to choose s N.

IN TOTAL


48+12

TOTAL NUMBERS 90

60/96=5/8 is the ANSWER