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# If P, Q, and R are positive integers, what is the remainder

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Manager
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If P, Q, and R are positive integers, what is the remainder [#permalink]

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27 Oct 2012, 19:48
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If P, Q, and R are positive integers, what is the remainder when R – Q is divided by 3?

(1) R = P^3
(2) Q = (P – 2)^3
[Reveal] Spoiler: OA

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Last edited by Bunuel on 29 Oct 2012, 05:35, edited 1 time in total.
Renamed the topic.

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27 Oct 2012, 23:14
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Pansi wrote:
If P, Q, and R are positive integers, what is the remainder when R – Q is divided by 3?

(1) R = P^3
(2) Q = (P – 2)^3

2) No information about R. Insufficient

1 & 2 together. $$R-Q = P^3 - (P-2)^3$$

So $$P^3$$gets cancelled out and we only have multiples of 3 and a constant remaining.

Sufficient

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29 Oct 2012, 05:37
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If P, Q, and R are positive integers, what is the remainder when R – Q is divided by 3?

(1) R = P^3. No info about Q. Not sufficient.
(2) Q = (P – 2)^3. No info about R. Not sufficient.

(1)+(2) Important tip: x^3-y^3 can be factored as follows: $$x^3-y^3=(x-y)(x^2+xy+y^2)$$. Apply this factoring to $$R-Q$$ --> $$R-Q=P^3-(P-2)^3=(P-P+2)(P^2+P^2-2P+P^2-4P+4)=2(3P^2-6P+4)=6c^2-12c+8=6(c^2-2c+1)+2$$ --> remainder upon division this expression by 3 is 2. Sufficient.

Similar questions to practice:
if-a-b-and-c-are-positive-integers-what-is-the-remainder-140495.html
if-a-b-and-c-are-positive-integers-what-is-the-remainder-127028.html

Hope it helps.
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Re: If P, Q, and R are positive integers, what is the remainder [#permalink]

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10 Aug 2013, 08:39
Hi folks,

Both are insufficient without any doubts checking for option (C).

P=3 , P-2 =1 27-1 = 26/3 =2

P=4 , P-2 = 2 64-8 = 56/3 =2

P=5 , P-2 = 3 125-27 , Rem=2

P=6, P-2 = 4 Rem =2

Hence (C)
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Re: If P, Q, and R are positive integers, what is the remainder [#permalink]

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24 Oct 2015, 13:05
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Re: If P, Q, and R are positive integers, what is the remainder [#permalink]

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20 Jul 2017, 18:00
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If P, Q, and R are positive integers, what is the remainder   [#permalink] 20 Jul 2017, 18:00
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