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as it is a choice problem pick a value of p. 11 seems the most reasonable

11^6-1 = 1771560

this number is divisible by 7.
this number is divisible by 8 since 560 is divisible by 8
this number is divisible by 9 since 1+7+7+1+5+6+0=27 which is divisible by 9

implies this number is divisible by 7*8*9

PS:
For checking for divisibility by 7. There are many methods for checking
I use this one. From the number work from right to left multiply by 1,2,3,-1,-2,-3,1,2,3,.... and add the result

for this case 1*0 + 3*6 +2*5 - 1*1 - 3*7-2*7+1*1= -7 which is divisible by 7.

Also Dan have proposed a method in an earlier post.

I don't know what's the logic or pattern here, but for different primes, P^6 - 1 sometimes is and sometimes is not divisible by one or more of the choices, but is surely divisble by choice #4 (7*8*9 = 504). This is true for all primes above 7. It does not work for primes 2 or 3.

I have this question but not the OA. Maybe it involves algorithm and is beyond the GMAT scope (and maybe not) but am curious.

In the question stem, 7 is the lower limit, and it is also the lower limit in the 'correct' answer of 7*8*9.

Say n = number is 3, then 3^3 = 27 which is 1 less than a multiple of 7. Say n = 2, n^3 is 1 more than a multiple of 7.

therefore, for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.

then all primes fit in (since primes have no multiples, not to mention 7), except 7 (since it is a multiple of 7), and this is why it is excluded in the question.

So, the divisor must be a multiple of 7, so answer 3 or 4, but since the question limits it for primes above 7, the other multiples have to be larger than 7 too.