BTGmoderatorDC wrote:If n is an integer, which of the following must be divisible by 3?
A) n^3 - 4n
B) n^3 + 4n
C) n^2 +1
D) n^2 -1
E) n^2 -4
OA A
Source: Manhattan Prep
Simplifying answer choice A, we have:
n^3 - 4n = n(n^2 - 4) = n(n + 2)(n - 2) = (n - 2)(n)(n + 2)
This is a product of 3 consecutive odd integers or 3 consecutive even integers, and since one of the factors must be a multiple of 3, we see that no matter what n is, the product will always be a multiple of 3. For example,
When n is 1, we have:
-1 x 1 x 3
When n is 2, we have:
0 x 2 x 4
When n is 3, we have:
1 x 3 x 5
Thus, answer choice A will always be divisible by 3.
Alternate Solution:
Let's pick some convenient values for n:
If n = 0, then 0^2 - 1 = -1; thus, we can eliminate answer choice D. Further, 0^2 - 4 = -4; thus, we can eliminate answer choice E.
If n = 1, then 1^3 + 4(1) = 5; thus, we can eliminate answer choice B. Further, 1^2 + 1 = 2, so we can eliminate answer choice C.
The only remaining answer choice is A.
Answer: A
