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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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Updated on: 04 Oct 2017, 03:10
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67% (01:35) correct 33% (01:29) wrong based on 308 sessions

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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

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Originally posted by vivek6199 on 04 Oct 2017, 03:08.
Last edited by Bunuel on 04 Oct 2017, 03:10, edited 1 time in total.
Renamed the topic.
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Re: If n is an integer, which of the following must be divisible by 3?  [#permalink]

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04 Oct 2017, 03:13
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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.

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

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04 Oct 2017, 03:44
2
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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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

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.

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

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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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10 Oct 2017, 10:25
2
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.

Check below for more:
ALL YOU NEED FOR QUANT ! ! !

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

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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]

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26 Jul 2018, 10:22
Bunuel wrote:
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.

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 + $$a^2+b^2$$
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Re: If n is an integer, which of the following must be divisible by 3?  [#permalink]

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26 Jul 2018, 10:35
1
dave13 wrote:
Bunuel wrote:
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.

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 + $$a^2+b^2$$

Because n^2 + 4 does not equal to (n + 2)(n - 2).
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If n is an integer, which of the following must be divisible by 3?  [#permalink]

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26 Jul 2018, 10:46
2
dave13 wrote:
Bunuel wrote:
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.

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 + $$a^2+b^2$$

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

I have merely expanded on Bunuel 's option!
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Re: If n is an integer, which of the following must be divisible by 3?  [#permalink]

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26 Jul 2018, 21:20
1
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

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.
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Re: If n is an integer, which of the following must be divisible by 3? &nbs [#permalink] 26 Jul 2018, 21:20
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