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# If n is a positive integer, is n^3 + 4n^2 – 5n divisible by

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If n is a positive integer, is n^3 + 4n^2 – 5n divisible by  [#permalink]

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Updated on: 14 Jan 2013, 06:42
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Difficulty:

65% (hard)

Question Stats:

63% (02:40) correct 38% (02:25) wrong based on 154 sessions

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If n is a positive integer, is n^3 + 4n^2 - 5n divisible by 8 ?

(1) n = 4b + 1, where b is a positive integer.

(2) n^2 – n is divisible by 24.

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Originally posted by daviesj on 14 Jan 2013, 06:40.
Last edited by Bunuel on 14 Jan 2013, 06:42, edited 1 time in total.
Edited the question.
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Re: If n is a positive integer, is n^3 + 4n^2 – 5n divisible by  [#permalink]

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14 Jan 2013, 06:49
If n is a positive integer, is n^3 + 4n^2 - 5n divisible by 8 ?

$$n^3 + 4n^2 - 5n=n(n^2+4n-5)=n(n-1)(n+5)$$

(1) n = 4b + 1, where b is a positive integer --> $$n(n-1)(n+5)=(4b+1)(4b)(4b+6)=(4b+1)(8b)(2b+3)$$. Sufficient.

(2) n^2 – n is divisible by 24 --> n(n-1) is divisible by 24, so its also divisible by 8. Thus n(n-1)(n+5) is also divisible by 8. Sufficient.

Hope it's clear.
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Re: If n is a positive integer, is n^3 + 4n^2 – 5n divisible by  [#permalink]

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15 Jan 2013, 11:56
Hi Bunuel,

Can u plz xplain this step:

(4b+1)(4b)(4b+6)= (4b+1)(8b)(2b+3).Sufficient.

Thanks,
Shreeraj
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Re: If n is a positive integer, is n^3 + 4n^2 – 5n divisible by  [#permalink]

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15 Jan 2013, 12:08
shreerajp99 wrote:
Hi Bunuel,

Can u plz xplain this step:

(4b+1)(4b)(4b+6)= (4b+1)(8b)(2b+3).Sufficient.

Thanks,
Shreeraj

Sure.

Factor out 2 from 4b+6: (4b+1)(4b)(4b+6)=(4b+1)(4b)(2)(2b+3)=(4b+1)(8b)(2b+3).

Hope it's clear.
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Re: If n is a positive integer, is n^3 + 4n^2 – 5n divisible by  [#permalink]

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17 Jan 2014, 12:32
daviesj wrote:
If n is a positive integer, is n^3 + 4n^2 - 5n divisible by 8 ?

(1) n = 4b + 1, where b is a positive integer.

(2) n^2 – n is divisible by 24.

The only things to point out or add if you will are the following

Question stem = Factorizing one gets (n)(n+5)(n-1)

Statement 1

n = 4b+1

Replacing (4b+1)(4b+6)(4b)

Now (4b) and (4b+1) are consecutive integers

Therefore one of them must be odd and one even, obviously 4b is the even one

So 4b is a multiple of 4 and 4b +6 is even so we have a multiple of 8

Statement 2

n(n-1) divisible by 24 means that either n or n-1 is divisible by 3 and that one of them, the even one, will be divisible by 8

Since we have both terms in our initial factorization then YES it is a multiple of 8

Hence D

Is this clear?

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Cheers!
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Re: If n is a positive integer, is n^3 + 4n^2 – 5n divisible by  [#permalink]

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16 Sep 2018, 02:54
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Re: If n is a positive integer, is n^3 + 4n^2 – 5n divisible by   [#permalink] 16 Sep 2018, 02:54
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