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If a, b, and c are positive integers, what is the remainder

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thevenus
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If a, b, and c are positive integers, what is the remainder [#permalink] New post   10 Oct 2012, 14:56
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If a, b, and c are positive integers, what is the remainder when a – b is divided by 6?

(1) a = c^3
(2) b = (c – 2)^3
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Re: If a, b, and c are positive integers, what is the remainder [#permalink] New post   10 Oct 2012, 15:15
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thevenus wrote:
If a, b, and c are positive integers, what is the remainder when a – b is divided by 6?

(1) a = c^3
(2) b = (c – 2)^3

Manhattan GMAT weekly challenge (2nd week, Oct,2012)


If a, b, and c are positive integers, what is the remainder when a – b is divided by 6?

(1) a = c^3. No info about b. Not sufficient.
(2) b = (c – 2)^3. No info about b. 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 \(a-b\) --> \(a-b=c^3-(c-2)^3=(c-c+2)(c^2+c^2-2c+c^2-4c+4)=2(3c^2-6c+4)=6c^2-12c+8=6(c^2-2c+1)+2\) --> remainder upon division this expression by 6 is 2. Sufficient.

Answer: C.

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Re: If a, b, and c are positive integers, what is the remainder [#permalink] New post   10 Oct 2012, 15:33
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thevenus wrote:
If a, b, and c are positive integers, what is the remainder when a – b is divided by 6?

(1) a = c^3
(2) b = (c – 2)^3

Manhattan GMAT weekly challenge (2nd week, Oct,2012)


Obviously, neither (1), nor (2) alone is sufficient.

Integers, when divided by 6 leave remainders 0, 1, 2, 3, 4, or 5.
When raised to the third power, the remainders stay the same 0, 1, 2, 3, 4, or 5:
\(6^3\)- remainder 0
\(1^3\) - remainder 1
\(2^3=8=6+2\) - remainder 2
\(3^3=27=24+3\) - remainder 3
\(4^3=64=60+4\) - remainder 4
\(5^3=125=120+5\) - remainder 5

Therefore, when subtracting the cubes of two integers two units apart, the difference will leave a remainder of 2 when divided by 6:
\(2-0=3-1=4-2=5-2\) and \(0-4=1-5=-4=-6+2.\)
Remainders repeat themselves cyclically \(0,1,2,3,4,5,0,1,2,3,4,5...\)

Answer C.
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Re: If a, b, and c are positive integers, what is the remainder [#permalink] New post   17 Aug 2013, 02:25
Would it make sense to pick numbers?

\(c=3\)
\(a = c^3 = 27\)
\(b=(c-2)^3 = 1\) (I know 1 is a bit problematic sometimes..)

\(a-b = 27-1 = 26\)
\(26/6 = 4 * 6 + 2\)

same goes for c = 4 ...
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Re: If a, b, and c are positive integers, what is the remainder [#permalink] New post   30 Apr 2024, 18:59
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Re: If a, b, and c are positive integers, what is the remainder [#permalink]
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