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if x and y are integers

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by ct18 » Thu Apr 26, 2012 8:00 pm
if x and y are integer, what is the remainder when x^2 and 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


OA: C

how do you solve this problem?
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by Shalabh's Quants » Fri Apr 27, 2012 12:21 am
ct18 wrote:if x and y are integer, what is the remainder when x^2 and 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


OA: C

how do you solve this problem?
Well looking the question, statements and answer, perhaps the question stem should be

if x and y are integer, what is the remainder when (x^2 - y^2) is divided by 5? . If it is so then...

We wish to find out remainder of (x^2 - y^2)/5.

It can be written as (x^2 - y^2)/5 = (x-y).(x+y)/5.

As remainders are multiplicative so remainder of (x^2 - y^2)/5 = remainder of [(x - y)/5 * (x + y)/5].

Statement 1...

Insuff. as it gives information about (x - y)/5 only.

Statement 2...

Insuff. as it gives information about (x + y)/5 only.

Together we can say

remainder of (x^2 - y^2)/5 = remainder of [(x - y)/5 * (x + y)/5] = 1 * 2 = 2.

Ans C.
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by GMATGuruNY » Fri Apr 27, 2012 3:43 am
ct18 wrote:if x and y are integer, what is the remainder when x^2 and 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


OA: C
Question rephrased: What is the remainder when (x+y)(x-y) is divided by 5?

Statement 1: When x-y is divided by 5, the remainder is 1.
In other words:
x-y = 5k + 1, where k≥0.
No information about x+y.
INSUFFICIENT.

Statement 2: When x+y is divided by 5, the remainder is 2.
In other words:
x+y = 5m + 2, where m≥0.
No information about x-y.
INSUFFICIENT.

Statements 1 and 2 combined:
(x-y)(x+y) = (5k+1)(5m+2) = 25km + 10k + 5m + 2.
The first three terms (25km, 10k, and 5m) are all multiples of 5.
Thus, when the entire expression is divided by 5, the remainder will be the last term:
r=2.
SUFFICIENT.

The correct answer is C.
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by ronnie1985 » Fri Apr 27, 2012 8:12 am
x-y = 5m+1
x+y = 5n+2

Adding and Subtracting.

2x = 5m+5n+3
2y = 5n-5m+1

m+n can be odd or even represented by 2k+1 or 2k
then 2x = 5(2k+1)+3 or 5(2k)+3
x = 5k+4 5k+1.5. Since x is integer the second option is ruled out.
x = 5k+4

Similarly for y.

hence (C) is the answer.
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by Goal750 » Fri Aug 10, 2012 7:44 pm
hi.
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by lazarogb » Sat Nov 24, 2012 11:57 am
GMATGuruNY wrote:
ct18 wrote:if x and y are integer, what is the remainder when x^2 and 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


OA: C
Question rephrased: What is the remainder when (x+y)(x-y) is divided by 5?

Statement 1: When x-y is divided by 5, the remainder is 1.
In other words:
x-y = 5k + 1, where k≥0.
No information about x+y.
INSUFFICIENT.

Statement 2: When x+y is divided by 5, the remainder is 2.
In other words:
x+y = 5m + 2, where m≥0.
No information about x-y.
INSUFFICIENT.

Statements 1 and 2 combined:
(x-y)(x+y) = (5k+1)(5m+2) = 25km + 10k + 5m + 2.
The first three terms (25km, 10k, and 5m) are all multiples of 5.
Thus, when the entire expression is divided by 5, the remainder will be the last term:
r=2.
SUFFICIENT.

The correct answer is C.
Hello Mitch,

Why did you rewrite x^2+y^2 into (x+y)(x-y)?

Because normally for us to be able to rewrite it like that we would need a difference of squares and not a sum. Maybe the post did not specify that the squares were being added or perhaps I am missing something.
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by GMATGuruNY » Sat Nov 24, 2012 12:50 pm
lazarogb wrote: Hello Mitch,

Why did you rewrite x^2+y^2 into (x+y)(x-y)?

Because normally for us to be able to rewrite it like that we would need a difference of squares and not a sum. Maybe the post did not specify that the squares were being added or perhaps I am missing something.
The wording of the question stem is vague: x² AND y² has no clear mathematical meaning.
The solution that I posted above answers the following question:
What is the remainder when x²-y² is divided by 5?
It's possible that the question stem intends to ask the following:
What is the remainder when x²+y² is divided by 5?
For the latter question, I posted a solution here:
https://www.beatthegmat.com/ds-remainder ... 41041.html
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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by lazarogb » Sat Nov 24, 2012 12:56 pm
GMATGuruNY wrote:
lazarogb wrote: Hello Mitch,

Why did you rewrite x^2+y^2 into (x+y)(x-y)?

Because normally for us to be able to rewrite it like that we would need a difference of squares and not a sum. Maybe the post did not specify that the squares were being added or perhaps I am missing something.
The wording of the question stem is vague: x² AND y² has no clear mathematical meaning.
The solution that I posted above answers the following question:
What is the remainder when x²-y² is divided by 5?
It's possible that the question stem intends to ask the following:
What is the remainder when x²+y² is divided by 5?
For the latter question, I posted a solution here:
https://www.beatthegmat.com/ds-remainder ... 41041.html
Yes I just saw that post!

I think its a hard question considering a 2 min time limit :S

Thanks Mitch!
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