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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]

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27 Jan 2013, 21:53
Bunuel wrote:
vikram4689 wrote:
Both are correct examples but the one i gave does not follow what you mentioned

You said since 2*P is divisible by 5, therefore P is divisible by 5... This is not correct P=5/2

Not so. Your example is NOT correct.

I'm saying that if p is an integer and 2p is divisible by 5 then p must be divisible by 5. Now, check your example, is it correct?

It should be: 2*5 is divisible by 5 (p=5) so 5 (p) is divisible by 5.

Hope it's clear.

I believe vikram4689 is correct. If we know that 2P is divisible by 5, then we only know that P is divisible by 2.5.

Let,
P=x^2+y^2 and
N=5a^2+2a+5b^2+4b+1

Therefore, we have already proven:

2P=5N
P=(5/2)N

The only way for P to be divisible by 5 is if N is even (contains a factor of 2). Unfortunately, we can't factor out a 2 from N. Therefore, whether or not N is even depends on the values of A and B.

By looking at the equation of N, we can see that it will be even only if A and B are not both positive or both negative.

It works out that the equations (x-y)=5a+1 and (x+y)=5b+2 can both be satisfied only when A and B are not both positive or both negative. This is why Bunuel's incorrect assumption still works in this case.

There is no way I could have figured all of this out in 2 or 3 minutes though.

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

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27 Jan 2013, 22:32
matt8 wrote:
Bunuel wrote:
vikram4689 wrote:
Both are correct examples but the one i gave does not follow what you mentioned

You said since 2*P is divisible by 5, therefore P is divisible by 5... This is not correct P=5/2

Not so. Your example is NOT correct.

I'm saying that if p is an integer and 2p is divisible by 5 then p must be divisible by 5. Now, check your example, is it correct?

It should be: 2*5 is divisible by 5 (p=5) so 5 (p) is divisible by 5.

Hope it's clear.

I believe vikram4689 is correct. If we know that 2P is divisible by 5, then we only know that P is divisible by 2.5.

Let,
P=x^2+y^2 and
N=5a^2+2a+5b^2+4b+1

Therefore, we have already proven:

2P=5N
P=(5/2)N

The only way for P to be divisible by 5 is if N is even (contains a factor of 2). Unfortunately, we can't factor out a 2 from N. Therefore, whether or not N is even depends on the values of A and B.

By looking at the equation of N, we can see that it will be even only if A and B are not both positive or both negative.

It works out that the equations (x-y)=5a+1 and (x+y)=5b+2 can both be satisfied only when A and B are not both positive or both negative. This is why Bunuel's incorrect assumption still works in this case.

There is no way I could have figured all of this out in 2 or 3 minutes though.

Think again - It's not 'Bunuel's incorrect assumption'. It is given that x and y are integers so, given that P=x^2+y^2, P will be a positive integer.
Now that we know that P is an integer, if 2P is divisible by 5, P must be divisible by 5.

Given that a and b are positive integers and that 'a' is divisible by 'b', a MUST have b as a factor. It's a basic mathematical fact. When I say, 'a' is divisible by 'b', I mean 'a' is a multiple of 'b' which means 'b' is a factor of 'a'. Try some examples to convince yourself.

This statement has far reaching implications.
Think of questions like: Is $$2^{100}$$ divisible by 3? I will say 'no' without a thought.
Is $$2^{10}*3^{100}*5$$ divisible by 7? Again, absolutely no!

To reiterate, if 'a' is to be divisible by 'b', a must have b as a factor. (talking about positive integers)
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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28 Jan 2013, 22:39
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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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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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30 Jan 2013, 22:48
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.

Answer: C.

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]

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05 Jul 2013, 02:44
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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26 Mar 2014, 22:26
from HTale's post on 17 May 2012: "You know that 2(x^2+y^2)= (x-y)^2 + (x+y)^2."

Please help: how does 2(x^2+y^2) factor to (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]

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13 Nov 2015, 15:11
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.

Answer: C.

Hope it's clear.

Question for Bunuel. Theoretically, could you have used only q or p to represent the qoutient in this example. Wouldn't they represent the same whole factor of times that 5 can go into each statement?

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If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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14 Dec 2015, 07:03
IMO this question is a walk in the park if we know the following rules:

You can always ignore the original number for remainder calculation. If you know a/n gives remainder p and b/n gives remainder of q then-
i) remainder of (a+b)/n = remainder of (p+q)/n
ii) remainder of (a-b)/n = remainder of (p-q)/n
iii) remainder of (a^2+b^2)/n = remainder of (p^2+q^2)/n
iv) remainder of (a^2-b^2)/n = remainder of (p^2-q^2)/n
v) remainder of (a*b)/n = remainder of (p*q)/n
vi) remainder of (a/b)/n = remainder of (p/q)/n
.
.
.
etc

But Im not quite sure about the accuracy of these rules.

So Moderators and Math Experts:

How do the above rules look neat or preposterous ?;) Any corrections, additions, exceptions? Thanks
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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23 Apr 2017, 01:10
kt750 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

(2) When x+y is divided by 5, the remainder is 2

Hi,

Clearly, statement 1 and 2 individually are not sufficient.

(1) When x-y is divided by 5, the remainder is 1

=> $$(x-y)^{2}$$ divided by 5, the remainder is 1. => $$x^{2} + y^{2} - 2xy = 5k + 1$$ ---(*)

(2) When x+y is divided by 5, the remainder is 2

=> $$(x+y)^{2}$$ divided by 5, the remainder is 4. => $$x^{2} + y^{2} + 2xy = 5m + 4$$ ---(**)

Add equation (*) and (**), we have following:

$$2*(x^{2} + y^{2}) = 5(k+m) + 5$$

Hence, when $$x^2 + y^2$$ is divided by 5, the remainder is 0.

Sufficient. Answer(C)

Thanks.

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

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18 Jun 2017, 13:32
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.

Answer: C.

Hope it's clear.

as usual a very good explanation

but how do I recognise when I have square the terms ??
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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16 Nov 2017, 17:06
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.

Answer: C.

Hope it's clear.

Hi Bunuel,

When you squared each term, why did you make X=5 and Y=1 in the first expression and X=5 and Y=2 in the second expression? I don't understand what indicated that.

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

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16 Nov 2017, 20:43
bkastan wrote:
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.

Answer: C.

Hope it's clear.

Hi Bunuel,

When you squared each term, why did you make X=5 and Y=1 in the first expression and X=5 and Y=2 in the second expression? I don't understand what indicated that.

Not following you... Which part are you talking about?
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

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16 Nov 2017, 20:45
Bunuel wrote:
bkastan wrote:
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.

Answer: C.

Hope it's clear.

Hi Bunuel,

When you squared each term, why did you make X=5 and Y=1 in the first expression and X=5 and Y=2 in the second expression? I don't understand what indicated that.

Not following you... Which part are you talking about?

How did you get the following: (1)+(2) Square both expressions: x2−2xy+y2=25q2+10q+1x2−2xy+y2=25q2+10q+1 and x2+2xy+y2=25p2+20p+4x2+2xy+y2=25p2+20p+4 --> add them up: 2(x2+y2)=5(5q2+2q+5p2+4p+1)2(x2+y2)=5(5q2+2q+5p2+4p+1) --> so 2(x2+y2)2(x2+y2)

How did x^2 turn into 25Q^2?

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

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16 Nov 2017, 21:55
Bunuel wrote:
bkastan wrote:
How did you get the following: (1)+(2) Square both expressions: x2−2xy+y2=25q2+10q+1x2−2xy+y2=25q2+10q+1 and x2+2xy+y2=25p2+20p+4x2+2xy+y2=25p2+20p+4 --> add them up: 2(x2+y2)=5(5q2+2q+5p2+4p+1)2(x2+y2)=5(5q2+2q+5p2+4p+1) --> so 2(x2+y2)2(x2+y2)

How did x^2 turn into 25Q^2?

$$x-y=5q+1$$ --> $$(x-y)^2=(5q+1)^2=25q^2+10q+1$$

$$x+y=5p+2$$ --> $$(x+y)^2=(5p+2)^2=25p^2+20p+4$$

Thank you so much. You're so smart!!

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

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17 Nov 2017, 00:13
kt750 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

(2) When x+y is divided by 5, the remainder is 2

In such questions for me it is more easier to solve with actual numbers

by seeing the question and reading the options you can easily sense that option A, B, D are wrong (choice will most probably be in between C & E)

Identify numbers for which conditions support
(1) x,y could be 7,6 then remainder will be 0
x,y could be 8,7 then remainder will be 3..not sufficient

(2)x,y could be 8,4 then remainder will be 0
x,y could be 7,5 then remainder will be 4..not sufficient

However on combining we can see that when x,y is 9,8 it passes both the conditions and the remainder is 0
We can check as well
when x,y is 4,3 it passes both the conditions and the remainder is 0
Hence C
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Re: If x and y are integer, what is the remainder when x^2 + y^2   [#permalink] 17 Nov 2017, 00:13

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