If n is an integer greater than 6, which of the following must be divisible by 3?
Choose the correct option.
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)
The product of 3 numbers to be divisible by 3 at least one of them must be divisible by 3. So, to ensure that the product of 3 integers shown is divisible by 3 all 3 numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3. We should have something like n(n+1)(n+2) (for example: if n divided by 3 yields remainder of 1, then n+1 yields remainder of 2 and n+2 yields remainder of 0, thus it's divisible by 3 OR if n divided by 3 yields remainder of 2, then n+2 yields remainder of 1 and n+1 yields remainder of 0, thus it's divisible by 3).   Only option A satisfies this, because n(n+1)(n−4)=n(n+1)(n−6+2) and n−6has the same remainder as nn upon division by 3 thus we can replace it by n.   N(N+1)(N-4)nswer: N(N+1)(N-4).
<result>A</result>
If n is an integer greater than 6, which of the following must be divisible by 3?