For those that need a visual guide, I found the answer by creating a simple matrix chart for the first 10 numbers. On the GMAT, whenever you see number property questions that involve a range of numbers, you know you are looking for the magic pattern in the set. So.....
n=1 1x2x3 = 6 <--- not divisible by 8
n=2 2x3x4 = 24 <----- divisible by 8
n=3 3x4x5 = 60 <-- not divisible by 8
n=4 4x5x6 = 120 <---- divisible by 8
and so on... What you immediately notice after the first few is that all the even numbers are divisible by 8 since 8 is really 2^3. Anytime you have a number with 3 factors of 2, it is divisible by 8. The real trick of this question is not stopping at the even numbers. You are suppose to ask yourself whether if there are any odd numbers when multiplied by the next two integers larger than it will be divisible by 8.
We know multiples of 8 such as 16, 24, 32, 40, 48, 56, etc, all the way to 96 (thats all we are concerned about because thats our set range) are divisible by 8 and that odd number n's such as 15 (15x16x17), 23 (23x24x26) will include an 8 multiple in the equation. There are 12 of these Odd numbers if you list them out. Unfortunately, you will be wasting a lot of time!!! You need to instead just realize that there are odd numbers that can fit the description and recognize that these odd numbers always have number that is a multiple of 4 as part equation. Makes sense since if an odd number x a multiple of 4 x an even number will always give you a number that includes 3 factors of 2. ( any number x 2 x 2 x 2 = divisible by 8).
Now heres the crazy hard part you are suppose to figure out in less than 10 seconds. Since there are 96 numbers, you divide it by 4 which gives you 24 total numbers divisible by 4, but only take half of them because the other half are even numbers you have already counted in the first part.
48 + 12 = 60 60 out of 96 = 5/8
n=1 1x2x3 = 6 <--- not divisible by 8
n=2 2x3x4 = 24 <----- divisible by 8
n=3 3x4x5 = 60 <-- not divisible by 8
n=4 4x5x6 = 120 <---- divisible by 8
and so on... What you immediately notice after the first few is that all the even numbers are divisible by 8 since 8 is really 2^3. Anytime you have a number with 3 factors of 2, it is divisible by 8. The real trick of this question is not stopping at the even numbers. You are suppose to ask yourself whether if there are any odd numbers when multiplied by the next two integers larger than it will be divisible by 8.
We know multiples of 8 such as 16, 24, 32, 40, 48, 56, etc, all the way to 96 (thats all we are concerned about because thats our set range) are divisible by 8 and that odd number n's such as 15 (15x16x17), 23 (23x24x26) will include an 8 multiple in the equation. There are 12 of these Odd numbers if you list them out. Unfortunately, you will be wasting a lot of time!!! You need to instead just realize that there are odd numbers that can fit the description and recognize that these odd numbers always have number that is a multiple of 4 as part equation. Makes sense since if an odd number x a multiple of 4 x an even number will always give you a number that includes 3 factors of 2. ( any number x 2 x 2 x 2 = divisible by 8).
Now heres the crazy hard part you are suppose to figure out in less than 10 seconds. Since there are 96 numbers, you divide it by 4 which gives you 24 total numbers divisible by 4, but only take half of them because the other half are even numbers you have already counted in the first part.
48 + 12 = 60 60 out of 96 = 5/8
