# If n is an integer greater than 6…

## Solution:

To determine which answer choice MUST be divisible by 3 we need to determine answer choice(s) in which there is at least one factor that is divisible by 3. Although there are conceptual ways to look at this question, this can more easily be completed by using simple substitution. We are told that n is an integer greater than 6, so let’s let n = 7:

A) n(n+1)(n-4)

7 x 8 x 3

This IS divisible by 3. Thus, A could be the answer.

B) n(n+2)(n-1)

7 x 9 x 8

This IS divisible by 3. Thus, B could be the answer.

C) n(n+3)(n-5)

7 x 10 x 2

This IS NOT divisible by 3. Thus, C cannot be the answer.

D) n(n+4)(n-2)

7 x 11 x 5

This IS NOT divisible by 3. Thus, D cannot be the answer.

E) n(n+5)(n-6)

7 x 12 x 1

This IS divisible by 3. Thus, E could be the answer.

Since answer choices A, B, and E all are divisible by 3 when n is 7, we choose another convenient value for n and determine which of the choices are still divisible by 3. Letting n = 8 we have:

A) n(n+1)(n-4)

8 x 9 x 5

This IS divisible by 3. Thus, A could be the answer.

B) n(n+2)(n-1)

8 x 10 x 7

This IS NOT divisible by 3. Thus, B cannot be the answer.

E) n(n+5)(n-6)

8 x 13 x 2

This IS NOT divisible by 3. Thus, E cannot be the answer.

We see that ONLY answer choice A is divisible by 3 in both instances, and, generally, answer choice A is divisible by 3 for ANY value of n greater than 6.

Note: Because every answer choices has the factor n, avoid using a multiple of 3 for n. Had we used a multiple of 3 for n, say 9, we would have had all answer choices divisible by 3, since 9 is already divisible by 3. By using 7 for n first, we were able to eliminate choices C and D. By next using 8 for n, we further eliminated choices B and E, leaving choice A as the only correct answer.