If n is an integer, which of the following must be divisible by 3?

Question

A) n^3 – 4n
B) n^3 + 4n
C) n^2 +1
D) n^2 -1
E) n^2 -4

Bunuel · Accepted Answer

Option A:  n^3 – 4n = n(n^2-4)=(n-2)n(n+2) .

(n-2)n(n+2) is the product of three consecutive odd or three consecutive even integers. In any case one of them must be a multiple of 3.

Answer: A.

Bunuel · Answer

ZERO:

1. 0 is an integer.

2. 0 is an even integer. An even number is an   integer   that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

3. 0 is neither positive nor negative integer (the only one of this kind).

4. 0 is divisible by EVERY integer except 0 itself.

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Hope it helps.

septwibowo · Answer

Solved in in one minute, plug in n=2 and n=3.

Bunuel  has offered the best algebraic approach.

ScottTargetTestPrep · Answer

Let’s simplify each answer choice:

A) n^3 – 4n

n(n^2 - 4) = n(n + 2)(n - 2) = (n - 2)(n)(n + 2)

We see that the expression above is a product of 3 consecutive even integers (if n is even) or the product of 3 consecutive odd integers (if n is odd). In either case, the product will always contain a prime factor of 3, so n^3 – 4n is always divisible by 3.

Alternate Solution:

If we take n = 1, we see that 1^3 + 4 = 5 is not a multiple of 3; thus, B cannot be the answer.

If we take n = 1, we see that 1^2 + 1 = 2 is not a multiple of 3; thus, C cannot be the answer.

If we take n = 3, we see that 3^2 - 1 = 8 is not a multiple of 3; thus, D cannot be the answer.

If we take n = 3, we see that n^2 - 4 = 5 is not a multiple of 3; thus, E cannot be the answer.

The only remaining choice is A.

Answer: A

zflodeen · Answer

I really struggled on this one and feel like I am missing something on this question. Isn't it possible for n to be zero as the question does not suggest otherwise? If you take the algebraic breakdown for A being (n-2)n(n+2), isn't it possible to be (-2)(0)(2)? I may be missing a math concept here, but I don't believe (-2)(0)(2) is divisible by 3.

Can someone help clear this up for me?

Thank you!

Can someone help clear this up for me?

Thank you!

zflodeen · Answer

Bunuel, thank you for clearing that up! I was missing point #4 and suspected that might be the case. I'm trying to dredge up all these old math concepts and appreciate all the insight on these forum posts.

dave13 · Answer

pushpitkc , if Bunuel says the product of three consecutive odd or three consecutive even integers. In any case one of them must be a multiple of 3. why cant I say the same of option B ?

n^3 + 4n = n(n^2+4)=(n+2)n(n+2) . here is the same product of three consecutive odd or three consecutive even integers. Bunuel used formula a^2-b^2 so I just changed the sign onto + a^2+b^2

Bunuel · Answer

Because n^2 + 4 does not equal to (n + 2)(n - 2).

pushpitkc · Answer

Hey dave13 (a^2 - 4) = (a^2 - 2^2) = (a + b)(a - b) Let a = 5 -> Left hand side is (5^2 - 2^2) = 25 - 4 = 21 | The right-hand side is (5 + 2)(5 - 2) = 7*3 = 21 But try it with your formula - you say (a^2 + 4) = (a + 2)(a + 2) Let a = 3 -> Left hand side is (3^2 + 2^2) = (9 + 4) = 13 | The right-hand side is (3 + 2)(3 + 2) = 5*5 = 25 Since these are not equal - your formula must be wrong

Hope this clears your confusion. I have merely expanded on Bunuel 's option!

Princ · Answer

OA: A

The product of three consecutive integers is divisible by 3.
Three consecutive integers be  n-1, n,n+1 
Product of these three consecutive integers :  (n-1)(n)(n+1)=(n^2-1)n=n^3-n 
 3n  is also divisible by 3.
 (n^3-n)-(3n)  i.e  n^3 – 4n  would also be divisible by 3.

Mansoor50 · Answer

how is (n-2)(n)(n+2) consecutive?

if we take (n-3)(n)(n+3), for n as 5 this gives us: 2 x 5 x 8 and as can be seen, is not divisible by 3. But this divisible by 3 if n is even

for (n-4)(n)(n+4) is divisible by 3 for even or odd n

KarishmaB · Answer

As far as divisibility by 3 is concerned,

(n - 2) is the same as (n + 1) because if (n - 2) is divisible by 3, then so is (n + 1) (which is just (n - 2 + 3)). If (n - 2) leaves a remainder of 1, so will (n + 1). If (n - 2) leaves a remainder of 2, so will (n + 1).

By the same concept, (n + 2) is the same as (n - 1)

So in effect, what we are looking at is this: (n - 1)*n*(n + 1)

Franzhel · Answer

I think this problem is wrongly classified as Distance/Rate problem. If someone could please change it

Bunuel · Answer

______________________
Edited the tags. Thank you.

bumpbot · Answer

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If n is an integer, which of the following must be divisible by 3?

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If n is an integer, which of the following must be divisible by 3?

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