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]

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]

31 Jan 2013, 12:06

rohitgupta86 wrote:

What about numbers such as (9 and 3) or (8 and 14) or (3 and 14) etc? By just taking numbers, can you be sure that the remainder will be 0 in each case? You will need to think of the logic behind it - you can either check for numbers and then look for logic or you can jump to the logic straight away.
(Statement 1 says that the remainder when x-y is divided by 5 will be 1, not that the difference between x and y is 1)

As a general rule, it is very hard to establish that the result will be the same in every case using number plugging. It is much easier to say that it will not be the same in every case using this strategy. So number plugging is best avoided in DS until and unless you feel that taking some easy numbers will show you that the result will not be the same in every case.
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Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink]

15 Feb 2013, 06:22

hard question

if 2*A is divided by 5
then A is divided by 5 because 2 is not divided by 5

I will follow.

great Bruno. thank you

Re: If x and y are integer, what is the remainder when x^2 + y^2 [#permalink] 15 Feb 2013, 06:22

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

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