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

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If a, b, and c are positive integers, what is the remainder  [#permalink]

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10 Oct 2012, 13:56
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71% (01:55) correct 29% (02:02) wrong based on 352 sessions

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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]

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10 Oct 2012, 14:15
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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.

Hope it helps.
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Re: If a, b, and c are positive integers, what is the remainder  [#permalink]

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10 Oct 2012, 14:33
1
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 $$0,1,2,3,4,5,0,1,2,3,4,5...$$

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Re: If a, b, and c are positive integers, what is the remainder  [#permalink]

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17 Aug 2013, 01:25
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]

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07 Sep 2018, 00:33
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Re: If a, b, and c are positive integers, what is the remainder &nbs [#permalink] 07 Sep 2018, 00:33
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