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# If n is a positive integer and if (n^3 - n)/(n+1) = 240, the

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If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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27 May 2012, 08:27
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If n is a positive integer and if (n^3 - n)/(n+1) = 240, then what is the value of n?

A. 12
B. 16
C. 17
D. 20
E. 48
[Reveal] Spoiler: OA
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If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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27 May 2012, 23:45
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macjas wrote:
If n is a positive integer and if (n^3 - n)/(n+1) = 240, then what is the value of n?

A. 12
B. 16
C. 17
D. 20
E. 48

$$\frac{n^3-n}{n+1}=240$$ --> $$\frac{n(n-1)(n+1)}{n+1}=240$$ --> $$n(n-1)=240$$ --> 240 is the product of two positive consecutive integers $$n-1$$ and $$n$$ --> $$n=16$$ (16*15=240).

Note that one other pair of consecutive integers satisfy $$n(n-1)=240$$: -15 and -16 but we are told that n is positive hence this solution is not valid.
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Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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26 Oct 2014, 22:33
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$$\frac{n^3 -n}{n+1} = 240$$

$$\frac{n(n+1)(n-1)}{n+1} = 240$$

$$n^2 - n - 240 = 0$$

$$n^2 - 16n + 15n - 240 = 0$$

n = 16

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If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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24 Nov 2015, 11:28
macjas wrote:
If n is a positive integer and if (n^3 - n)/(n+1) = 240, then what is the value of n?

A. 12
B. 16
C. 17
D. 20
E. 48

$$\frac{(n^3 - n)}{(n+1)}$$ = 240

$$\frac{(n^3 - n)}{(n+1)}$$ = $$(16 )(15)$$

$$\frac{n(n^2 - 1)}{(n+1)}$$ = $$(16 )(15)$$

$$\frac{n(n - 1)(n+1)}{(n+1)}$$= $$(16 )(15)$$

$$n(n - 1)$$ = $$(16 )(15)$$

So, $$n$$ = $$16$$

Hence answer is definitely (B)
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Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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29 Sep 2017, 09:30
macjas wrote:
If n is a positive integer and if (n^3 - n)/(n+1) = 240, then what is the value of n?

A. 12
B. 16
C. 17
D. 20
E. 48

We can simplify the given equation:

(n^3 - n)/(n + 1) = 240

n(n^2 - 1)/(n + 1) = 240

n(n + 1)(n - 1)/(n + 1) = 240

n(n - 1) = 240

n^2 - n - 240 = 0

(n - 16)(n + 15) = 0

n = 16 or n = -15

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Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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05 Oct 2017, 17:23
Bunuel wrote:
macjas wrote:
If n is a positive integer and if (n^3 - n)/(n+1) = 240, then what is the value of n?

A. 12
B. 16
C. 17
D. 20
E. 48

$$\frac{n^3-n}{n+1}=240$$ --> $$\frac{n(n-1)(n+1)}{n+1}=240$$ --> $$n(n-1)=240$$ --> 240 is the product of two positive consecutive integers $$n-1$$ and $$n$$ --> $$n=16$$ (16*15=240).

What is the significance of "If n is a positive integer" in this question? Would we not cancel (n+1) if n is not a positive integer?
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Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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05 Oct 2017, 20:07
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Bunuel wrote:
macjas wrote:
If n is a positive integer and if (n^3 - n)/(n+1) = 240, then what is the value of n?

A. 12
B. 16
C. 17
D. 20
E. 48

$$\frac{n^3-n}{n+1}=240$$ --> $$\frac{n(n-1)(n+1)}{n+1}=240$$ --> $$n(n-1)=240$$ --> 240 is the product of two positive consecutive integers $$n-1$$ and $$n$$ --> $$n=16$$ (16*15=240).

What is the significance of "If n is a positive integer" in this question? Would we not cancel (n+1) if n is not a positive integer?

We would but after getting $$n(n-1)=240$$ we wouldn't be able to use the logic in highlighted part because if it were given that n is an integer, then $$n-1$$ and $$n$$ might not be consecutive integers and we would be left with quadratics to solve. We'd get the same answer though.
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Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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05 Oct 2017, 22:05
What is the significance of "If n is a positive integer" in this question? Would we not cancel (n+1) if n is not a positive integer?[/quote]

We would but after getting $$n(n-1)=240$$ we wouldn't be able to use the logic in highlighted part because if it were given that n is an integer, then $$n-1$$ and $$n$$ might not be consecutive integers and we would be left with quadratics to solve. We'd get the same answer though.[/quote]

Hi Bunuel

Great work, you mentioned that we might not be able to get consecutive integers, but -16 and -15 are also consecutive integers and as you have mentioned we would arrive at the same solution. I would lean towards the isolation of a single solution as the significance of the positive integer constraint. As you are well aware of, this would probably be more significant in a DS yes or no or even a DS value question.

Thanks again for the excellent contributions.
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If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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05 Oct 2017, 22:10
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rulingbear wrote:
Great work, you mentioned that we might not be able to get consecutive integers, but -16 and -15 are also consecutive integers and as you have mentioned we would arrive at the same solution. I would lean towards the isolation of a single solution as the significance of the positive integer constraint. As you are well aware of, this would probably be more significant in a DS yes or no or even a DS value question.

Thanks again for the excellent contributions.

Notice that we are not only told that n is an integer but also that n is positive: " n is a positive integer".
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If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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05 Oct 2017, 22:27
Bunuel wrote:
We would but after getting $$n(n-1)=240$$ we wouldn't be able to use the logic in highlighted part because if it were given that n is an integer, then $$n-1$$ and $$n$$ might not be consecutive integers and we would be left with quadratics to solve. We'd get the same answer though.

I like the idea of plugging and not solving the quadratic equation...this saves a hell of time
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Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the [#permalink]

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05 Oct 2017, 23:13
[/quote]

What is the significance of "If n is a positive integer" in this question? Would we not cancel (n+1) if n is not a positive integer?[/quote]

We would but after getting $$n(n-1)=240$$ we wouldn't be able to use the logic in highlighted part because if it were given that n is an integer, then $$n-1$$ and $$n$$ might not be consecutive integers and we would be left with quadratics to solve. We'd get the same answer though.[/quote]

Thank you. This helps
Re: If n is a positive integer and if (n^3 - n)/(n+1) = 240, the   [#permalink] 05 Oct 2017, 23:13
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# If n is a positive integer and if (n^3 - n)/(n+1) = 240, the

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