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# If z is a positive integer and r is the remainder when z2 + 2z + 4 is

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Manager
Joined: 14 Sep 2015
Posts: 65
Location: India
GMAT 1: 700 Q45 V40
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If z is a positive integer and r is the remainder when z2 + 2z + 4 is  [#permalink]

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29 Jul 2017, 20:13
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Difficulty:

65% (hard)

Question Stats:

25% (02:13) correct 75% (02:08) wrong based on 8 sessions

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If z is a positive integer and r is the remainder when $$z^2$$ + 2z + 4 is divided by 8, what is the value of r?

(1) When $$(z-3)^2$$ is divided by 8, the remainder is 4
(2) When 2z is divided by 8, the remainder is 2

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Joined: 02 Aug 2009
Posts: 7686
Re: If z is a positive integer and r is the remainder when z2 + 2z + 4 is  [#permalink]

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29 Jul 2017, 20:54
niteshwaghray wrote:
If z is a positive integer and r is the remainder when $$z^2$$ + 2z + 4 is divided by 8, what is the value of r?

(1) When $$(z-3)^2$$ is divided by 8, the remainder is 4
(2) When 2z is divided by 8, the remainder is 2

Hi..

lets check the statements..

(1) When $$(z-3)^2$$ is divided by 8, the remainder is 4

so $$(z-3)^2=z^2-6z+9$$ is divided by 8, REMAINDER is 4...
this means $$z^2-6Z+9-4 = z^2-6z+5=(z-5)(z-1)$$ is div by 8..
For this z has to be of the form of 4k+1, where k is a non negative integer..
example k = 1, z=4k+1=4*1+1=5......(5-5)(5-1)=0 div by 8..
k=2, z=4*2+1=9......(9-5)(9-1)=4*8, div by 8..

substitute value of z in main equation..
$$z^2$$ + 2z + 4 .......$$(4k+1)^2+2(4k+1)+4=16k^2+8k+1+8k+2+4=16k^2+16k+7$$...
when this is div by 8 ONLY term 7 is left..
so remainder = 7

sufficient

(2) When 2z is divided by 8, the remainder is 2
so 2z=8a+2 or z=4a+1 where a is non negative integer..
now we have same value as in statement I, so again remainder will come as 7
sufficient

D

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THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

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Re: If z is a positive integer and r is the remainder when z2 + 2z + 4 is   [#permalink] 29 Jul 2017, 20:54
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