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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? [#permalink]

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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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 04 Oct 2017, 03:10, edited 1 time in total.
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Re: If n is an integer, which of the following must be divisible by 3? [#permalink]

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vivek6199 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


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.
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Re: If n is an integer, which of the following must be divisible by 3? [#permalink]

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New post 04 Oct 2017, 03:44
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vivek6199 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


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

Bunuel has offered the best algebraic approach.
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Re: If n is an integer, which of the following must be divisible by 3? [#permalink]

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New post 06 Oct 2017, 10:58
vivek6199 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



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
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Re: If n is an integer, which of the following must be divisible by 3? [#permalink]

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New post 10 Oct 2017, 10:17
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!

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Re: If n is an integer, which of the following must be divisible by 3? [#permalink]

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New post 10 Oct 2017, 10:25
zflodeen wrote:
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!


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.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If n is an integer, which of the following must be divisible by 3? [#permalink]

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New post 10 Oct 2017, 10:37
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.

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Re: If n is an integer, which of the following must be divisible by 3?   [#permalink] 10 Oct 2017, 10:37
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If n is an integer, which of the following must be divisible by 3?

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