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

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by M7MBA » Thu Oct 29, 2020 12:33 pm

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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

Source: Beat The GMAT — Problem Solving | Thu Oct 29, 2020 12:33 pm

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

by Scott@TargetTestPrep » Thu Nov 05, 2020 3:38 pm

M7MBA wrote: ↑
Thu Oct 29, 2020 12:33 pm
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

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