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Smita04
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If a, b, and c are positive integers, what is the remainder Updated on: 25 Jun 2014, 01:59
Originally posted by Smita04 on 03 Feb 2012, 21:45.
Last edited by Bunuel on 25 Jun 2014, 01:59, edited 2 times in total.
Edited the question and added the OA
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C
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73% (01:37) correct 27% (01:52) wrong based on 609 sessions

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If a, b, and c are positive integers, what is the remainder after b - a is divided by 3?

(1) a = c^3
(2) b = (c + 1)^3
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C
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If a, b, and c are positive integers, what is the remainder 04 Feb 2012, 03:55
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Smita04
If a , b , and c are positive integers, what is the remainder after b - a is divided by 3?

(1) a = c^3
(2) b = (c + 1)^3
Show more

If a , b , and c are positive integers, what is the remainder after b - a is divided by 3?

(1) a = c^3 --> no info about b. Not sufficient.
(2) b = (c + 1)^3 --> no info about b. Not sufficient.

When taken together you can go with algebraic approach or plug-in method:

Algebraic approach:

(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 \(b-a\) --> \(b-a=(c + 1)^3-c^3=(c+1-c)(c^2+2c+1+c^2+c+c^2)=3c^2+3c+1=3(c^2+c)+1\) --> remainder upon division this expression by 3 is 1. Sufficient.

Answer: C.

Plug-in method approach:

(1)+(2) try some numbers for a and b:
\(a=c^3=1\) --> \(b=(c+1)^3=8\) and --> \(b-a=(c + 1)^3-c^3=7\) --> remainder upon division 3 is 1;
\(a=c^3=8\) --> \(b=(c+1)^3=27\) and --> \(b-a=(c + 1)^3-c^3=19\) --> remainder upon division 3 is 1;
\(a=c^3=27\) --> \(b=(c+1)^3=64\) and --> \(b-a=(c + 1)^3-c^3=37\) --> remainder upon division 3 is 1;
\(a=c^3=64\) --> \(b=(c+1)^3=125\) and --> \(b-a=(c + 1)^3-c^3=61\) --> remainder upon division 3 is 1;
...
It seem that there is some kind of pattern and we can safely assume that in all other cases remainder will also be 1. Sufficient.

Answer: C.

Hope it helps.
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If a, b, and c are positive integers, what is the remainder 10 Oct 2012, 15:07
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(A) & (B) ruled out as we do not get complete info about a & b together, because in question we are supposed to find
remainder when (b-a) is divided by 3.

lets take both together;

Given:
a=c^3;b=(c+1)^3 ) ( for a,b,c all positive integer)
so lets check at three points;
@c=1, a=1,b=8 ; @ c=2, a=8, b= 27, @ c= 3 , a=27,b=64

8-1/3 R->1
27-8/3 R->1
64-27/3 R->1

Hence C is the answer
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If a, b, and c are positive integers, what is the remainder 10 Oct 2012, 16:01
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Bunuel
Smita04
If a , b , and c are positive integers, what is the remainder after b - a is divided by 3?

(1) a = c^3
(2) b = (c + 1)^3

If a , b , and c are positive integers, what is the remainder after b - a is divided by 3?

(1) a = c^3 --> no info about b. Not sufficient.
(2) b = (c + 1)^3 --> no info about b. Not sufficient.

When taken together you can go with algebraic approach or plug-in method:

Algebraic approach:

(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 \(b-a\) --> \(b-a=(c + 1)^3-c^3=(c+1-c)(c^2+2c+1+c^2+c+c^2)=3c^2+3c+1=3(c^2+c)+1\) --> remainder upon division this expression by 3 is 1. Sufficient.

Answer: C.

Plug-in method approach:

(1)+(2) try some numbers for a and b:
\(a=c^3=1\) --> \(b=(c+1)^3=8\) and --> \(b-a=(c + 1)^3-c^3=7\) --> remainder upon division 3 is 1;
\(a=c^3=8\) --> \(b=(c+1)^3=27\) and --> \(b-a=(c + 1)^3-c^3=19\) --> remainder upon division 3 is 1;
\(a=c^3=27\) --> \(b=(c+1)^3=64\) and --> \(b-a=(c + 1)^3-c^3=37\) --> remainder upon division 3 is 1;
\(a=c^3=64\) --> \(b=(c+1)^3=125\) and --> \(b-a=(c + 1)^3-c^3=61\) --> remainder upon division 3 is 1;
...
It seem that there is some kind of pattern and we can safely assume that in all other cases remainder will also be 1. Sufficient.

Answer: C.

Hope it helps.
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Integers, when divided by 3 can leave a remainder of \(0, 1, or 2.\)
Integers cubed, when divided by 3, will leave the same remainders, because \(0^3=0, 1^3=1,\) and \(2^3=8=6+2.\)

Therefore, when subtracting cubes of two consecutive integers, the result will always leave a remainder of 1:
the remainders repeat themselves cyclically \(0,1,2,0,1,2,...\), so \(1-0=2-1=1\) and \(\,\,0-2=-3+1.\)
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If a, b, and c are positive integers, what is the remainder 07 Oct 2022, 00:43
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Smita04
If a, b, and c are positive integers, what is the remainder after b - a is divided by 3?

(1) a = c^3
(2) b = (c + 1)^3
Show more
Solution:
Pre Analysis:
  • Given a, b, and c are positive integers
  • We are asked the remainder when b-a is divided by 3 i.e., we are asked the value of \((\frac{b-a}{3})_r\) or \((\frac{b}{3})_r-(\frac{a}{3})_r\)

Statement 1: \(a = c^3\)
  • No information on b
  • Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: \(b = (c + 1)^3\)
  • No information on a
  • Thus, statement 2 alone is also not sufficient and we can eliminate option B

Combining:
  • We have \((\frac{b-a}{3})_r\)
    \(⇒(\frac{(c+1)^3-c^3}{3})_r\)
    \(⇒(\frac{c^3+1+3c^2+3c-c^3}{3})_r\)
    \(⇒(\frac{1+3c^2+3c}{3})_r\)
    \(⇒(\frac{1}{3})_r+(\frac{3c(c+1)}{3})_r\)
    \(⇒1+0\)
  • Thus, after combining, we are able to get the answer

Hence the right answer is Option C
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