Divisibility problem

HG10 wrote:If n is an integer greater than 6, which of the following must be divisible by 3?

A. n(n+1)(n-4)
B. n(n+2)(n-1)
C. n(n+3)(n-5)
D. n(n+4)(n-2)
E. n(n+5)(n-6)

OA: A

Source: OG 12, #82

For the product of 3 numbers to be divisible by 3, at least one of them must be divisible by 3.
To check that the product of 3 integers is divisible by 3, all the 3 numbers should have different remainders on division by 3, which means that one of them should have a remainder of 1, other should have a reminder of 2 and the last one a remainder of 0.

We should have n(n + 1)(n + 2), because when divided by 3, if n gives a remainder of 1, then n + 1 gives a remainder of 2, and n + 2 gives a remainder of 0 OR if n gives a remainder of 2, then n + 2 gives a remainder of 1, and n + 1 gives a remainder of 0.

Only, answer choice A satisfies this, as n(n + 1)(n - 4) = n(n + 1)(n - 6 + 2) and n - 6 has the same remainder as when n is divided by 3. So, n can be replaced.

The correct answer is A.