If \(n\) is an integer greater than \(50,\) then the expression \((n^2-2n)(n^2-1)\) MUST be divisible by which of the

[M7MBA]

If \(n\) is an integer greater than \(50,\) then the expression \((n^2-2n)(n^2-1)\) MUST be divisible by which of the following?

I. 4
II. 6
III. 18

(A) I only
(B) II only
(C) I & II only
(D) II & III only
(E) I, II, and III

Answer: C

Source: Magoosh

[Scott@TargetTestPrep]

If \(n\) is an integer greater than \(50,\) then the expression \((n^2-2n)(n^2-1)\) MUST be divisible by which of the following?

I. 4
II. 6
III. 18

(A) I only
(B) II only
(C) I & II only
(D) II & III only
(E) I, II, and III

Answer: C

Solution:

Simplifying, we have:

n(n - 2)(n - 1)(n + 1) = (n - 2)(n - 1)(n)(n + 1)

We see that this is a product of 4 consecutive integers, which is always divisible by 4! = 24 (and therefore, any factors of 24). Since 4 and 6 are factors of 24, then (n^2 - 2n)(n^2 - 1) is divisible by 4 and 6. However, since 18 is not a factor of 24, (n^2 - 2n)(n^2 - 1) might not be divisible by 18.

Answer: C