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

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If P, Q, and R are positive integers, what is the remainder [#permalink] New post   Updated on: 29 Oct 2012, 05:35
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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
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C

Originally posted by Pansi on 27 Oct 2012, 19:48.
Last edited by Bunuel on 29 Oct 2012, 05:35, edited 1 time in total.
Renamed the topic.
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Re: Dividing cubicle [#permalink] New post   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


1)No information about Q. Insufficient
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

Answer is C.

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Re: Dividing cubicle [#permalink] New post   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.

Answer: C.

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

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Re: If P, Q, and R are positive integers, what is the remainder [#permalink] New post   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] New post   16 Jun 2018, 10:17
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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


Statement One Alone:

R = P^3

Since we don’t know anything about Q, statement one alone is not sufficient.

Statement Two Alone:

Q = (P - 2)^3

Since we don’t know anything about R, statement two alone is not sufficient.

Statements One and Two Together:

Since R = P^3

and

Q = (P - 2)^3

Q = (P^2 - 4P + 4)(P - 2)

Q = P^3 - 6P^2 + 12P - 8, R - Q

Q = 6P^2 - 12P + 8

Q = 3(2P^2 - 4P + 2) + 2.

We see that the remainder is 2 when R - Q = 3(2P^2 - 4P + 2) + 2 is divided by 3.

Answer: C
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Re: If P, Q, and R are positive integers, what is the remainder [#permalink] New post   07 Dec 2023, 08:26
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Re: If P, Q, and R are positive integers, what is the remainder [#permalink]
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