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If x and y are integer, what is the remainder when x^2 + y^2

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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]  28 Jan 2013, 21:39
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$$2*(x^2+y^2) = (x+y)^2+(x-y)^2$$
now,

x+y = 5k+2

x-y = 5m+1

$$(x+y)^2=25k^2+4+20k = 5(5k^2+4k)+4=5c+4$$

Similarly,
$$(x-y)^2=25m^2+1+10m=5(5m^2+2m)+1 = 5p+1$$

Hence, adding together the above 2 equations , we can get the remainder as;

5c+5p+5 = 5(p+c)+5= Multiple of 5. Hence the remainder is zero.

C.
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Last edited by mau5 on 05 Jul 2013, 01:47, edited 1 time in total.
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]  30 Jan 2013, 21:48
Bunuel wrote:
If x and y are integer, what is the remainder when x^2 + y^2 is divided by 5?

(1) When x-y is divided by 5, the remainder is 1 --> $$x-y=5q+1$$, so $$x-y$$ can be 1, 6, 11, ... Now, $$x=2$$ and $$y=1$$ ($$x-y=1$$) then $$x^2+y^2=5$$ and thus the remainder is 0, but if $$x=3$$ and $$y=2$$ ($$x-y=1$$) then $$x^2+y^2=13$$ and thus the remainder is 3. Not sufficient.

(2) When x+y is divided by 5, the remainder is 2 --> $$x+y=5p+2$$, so $$x+y$$ can be 2, 7, 12, ... Now, $$x=1$$ and $$y=1$$ ($$x+y=2$$) then $$x^2+y^2=2$$ and thus the remainder is 2, but if $$x=5$$ and $$y=2$$ ($$x+y=7$$) then $$x^2+y^2=29$$ and thus the remainder is 4. Not sufficient.

(1)+(2) Square both expressions: $$x^2-2xy+y^2=25q^2+10q+1$$ and $$x^2+2xy+y^2=25p^2+20p+4$$ --> add them up: $$2(x^2+y^2)=5(5q^2+2q+5p^2+4p+1)$$ --> so $$2(x^2+y^2)$$ is divisible by 5 (remainder 0), which means that so is $$x^2+y^2$$. Sufficient.

Hope it's clear.

HI...
y cant we just take examples...4 and 3 (14 and 13, 24 and 23)are the only nos. which will satisfy both the condition..
Hence, by checking cyclicity for 14 and 13 even...we can say that both the statements are reqd
wot say?
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]  05 Jul 2013, 01:44
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]  26 Mar 2014, 21:26
from HTale's post on 17 May 2012: "You know that 2(x^2+y^2)= (x-y)^2 + (x+y)^2."

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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]  28 Mar 2015, 09:04
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If x and y are integer, what is the remainder when x^2 + y^2   [#permalink] 28 Mar 2015, 09:04

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