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If n is an integer, which of the following must be divisible

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by BTGmoderatorDC » Fri Oct 19, 2018 7:20 pm

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Answers

A

B

C

D

E

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Difficulty—

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

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by GMATGuruNY » Sat Oct 20, 2018 2:44 am
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
Test n=0.

A --> n³ - 4n = 0³ - (4*0) = 0
B --> n³ + 4n = 0³ + (4*0) = 0
C --> n² + 1 = 0² + 1 = 1
D --> n² - 1 = 0² - 1 = -1
E --> n² - 4 = 0² - 4 = -4
Only the options in green yield a multiple of 3.
Eliminate C, D and E.

Test n=1 in the remaining answer choices.
A --> n³ - 4n = 1³ - (4*1) = -3
B --> n³ + 4n = 1³ + (4*1) = 5
Since B does not yield a multiple of 3, eliminate B.

The correct answer is A.
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by deloitte247 » Sun Oct 21, 2018 12:28 am
The product of 3 consecutive integers is divisible by 3
Three consecutive integers = n-1, n, n+1
product of these three consecutive numbers will be
$$\left(n-1\right)\left(n\right)\left(n+1\right)=\left(n^2-1\right)n$$
$$=n^3-n$$
3n is divisible by 3
$$\left(n^3-n\right)-\left(3n\right)$$
$$this\ means\ n^3-4n\ will\ \ also\ be\ divisible\ by\ 3$$
$$Answer\ is\ option\ A$$
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by Scott@TargetTestPrep » Thu Apr 04, 2019 5:35 pm
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

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