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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x and y are different positive integers, which of the following COU

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GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4226
If x and y are different positive integers, which of the following COU  [#permalink]

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1
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5 00:00

Difficulty:   95% (hard)

Question Stats: 27% (02:07) correct 73% (02:05) wrong based on 94 sessions

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If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

ASIDE: Many Integer Properties questions can be solved by identifying values that satisfy some given conditions. This question is intended to strengthen that skill.

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If x and y are different positive integers, which of the following COU  [#permalink]

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GMATPrepNow wrote:
If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

ASIDE: Many Integer Properties questions can be solved by identifying values that satisfy some given conditions. This question is intended to strengthen that skill.

First of all this question requires could be true answer.

i) could be.

This statement indicates that x has to be less than y.

x = 3

y = 4

3/4............3 is remainder here. Here the definition of remainder is needed.

So could be true.

ii) could be

iii) x = 2 and y = 1

2 + 1 / 2 = 2 *1 + 1

Remainder is 1

x - y

= 2 - 1 = 1.

Could be .

Actually there is no relationship between x and y is given. Thus any of them could be true in different ways.

Originally posted by KSBGC on 28 Oct 2018, 16:21.
Last edited by KSBGC on 28 Oct 2018, 18:23, edited 1 time in total.
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4226
Re: If x and y are different positive integers, which of the following COU  [#permalink]

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Top Contributor
Hi selim,

There's an error in one of your calculations.

Cheers,
Brent
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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GMATPrepNow wrote:
Hi selim,

There's an error in one of your calculations.

Cheers,
Brent

got it . thanks .
Senior Manager  P
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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All the options are possible as there is no exact relationship between integer x and y. So E is the answer.
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Joined: 02 Aug 2009
Posts: 8336
If x and y are different positive integers, which of the following COU  [#permalink]

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GMATPrepNow wrote:
If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

ASIDE: Many Integer Properties questions can be solved by identifying values that satisfy some given conditions. This question is intended to strengthen that skill.

Great question !! Brent..
let us see the question

i) When x is divided by y, the remainder is x
Whenever y>x, it will be true
example y=7 x=3, remainder is x or 3..possible

ii) When 2x is divided by y, the remainder is x
so let us make the equation..
$$2x=ny+x......ny=x$$, so x is multiple of y and remainder will always be 0... NOT possible

iii) When x+y is divided by x , the remainder is x-y
when x=y is divided by x, remainder will be y
Now can this y be equal to x-y....y=x-y... x=2y
so possible.. let y = 5, x=10.. remainder = 10-5=5=x-y....
so possible

(i) and (iii)

C
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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chetan2u wrote:
GMATPrepNow wrote:
If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

ASIDE: Many Integer Properties questions can be solved by identifying values that satisfy some given conditions. This question is intended to strengthen that skill.

Great question !! Brent..
let us see the question

i) When x is divided by y, the remainder is x
Whenever y>x, it will be true
example y=7 x=3, remainder is x or 3..possible

ii) When 2x is divided by y, the remainder is x
so let us make the equation..
$$2x=ny+x......ny=x$$, so x is multiple of y and remainder will always be 0... NOT possible

iii) When x+y is divided by x , the remainder is x-y
when x=y is divided by x, remainder will be y
Now can this y be equal to x-y....y=x-y... x=2y
so possible.. let y = 5, x=10.. remainder = 10-5=5=x-y....
so possible

(i) and (iii)

C

Hi Chetan, could you spare a few minutes to explain iii again? Thx Manager  B
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Posts: 157
Location: Pakistan
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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Great question !! Brent..
let us see the question

i) When x is divided by y, the remainder is x
Whenever y>x, it will be true
example y=7 x=3, remainder is x or 3..possible

ii) When 2x is divided by y, the remainder is x
so let us make the equation..
$$2x=ny+x......ny=x$$, so x is multiple of y and remainder will always be 0... NOT possible

iii) When x+y is divided by x , the remainder is x-y
when x=y is divided by x, remainder will be y
Now can this y be equal to x-y....y=x-y... x=2y
so possible.. let y = 5, x=10.. remainder = 10-5=5=x-y....
so possible

(i) and (iii)

C[/quote]

Hi chetan...

I dont understand your algebra for the III. can you elaborate please?

regards
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Posts: 73
Re: If x and y are different positive integers, which of the following COU  [#permalink]

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Is zero considered a positive integer? Because then X=0 would hold true for II, no?
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Posts: 4226
Re: If x and y are different positive integers, which of the following COU  [#permalink]

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hibobotamuss wrote:
Is zero considered a positive integer? Because then X=0 would hold true for II, no?

Zero is neither positive no negative.

Cheers,
Brent
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If x and y are different positive integers, which of the following COU  [#permalink]

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varun4s wrote:
All the options are possible as there is no exact relationship between integer x and y. So E is the answer.

If all of the statements are true, then for each statement, you should be able to specify values for x and y that satisfy the statement.

Hint: This is not possible for each statement (i.e., the correct answer is not E)

Cheers,
Brent
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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Algebraic approach to (iii)

x+y divided by x gives remainder of x-y

x+y-->px + x -y
2y-->px, where p is a positive integer. this means:
2y can be: x, 2x, 3x, 4x, ,,, and so on

2y eq x means y is x/2,,,,picking x as 4 gives y as 2 and thus
6/4 gives a remainder of 2 which is 4-2--> x-y

therefore, possible
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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chetan2u wrote:
GMATPrepNow wrote:
If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

ASIDE: Many Integer Properties questions can be solved by identifying values that satisfy some given conditions. This question is intended to strengthen that skill.

Great question !! Brent..
let us see the question

i) When x is divided by y, the remainder is x
Whenever y>x, it will be true
example y=7 x=3, remainder is x or 3..possible

ii) When 2x is divided by y, the remainder is x
so let us make the equation..
$$2x=ny+x......ny=x$$, so x is multiple of y and remainder will always be 0... NOT possible

iii) When x+y is divided by x , the remainder is x-y
when x=y is divided by x, remainder will be y
Now can this y be equal to x-y....y=x-y... x=2y
so possible.. let y = 5, x=10.. remainder = 10-5=5=x-y....
so possible

(i) and (iii)

C

Excuse me, how dividing 3/7 can give you a remainder of 3?
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Posts: 4226
Re: If x and y are different positive integers, which of the following COU  [#permalink]

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jorgetomas9 wrote:
Excuse me, how dividing 3/7 can give you a remainder of 3?

A few examples to set the mood:
25 divided by 8 equals 3 with remainder 1. From this result, we can write (3)(8) + 1 = 25
17 divided by 3 equals 5 with remainder 2. From this result, we can write (5)(3) + 2 = 17
64 divided by 10 equals 6 with remainder 4. From this result, we can write (6)(10) + 4 = 64
And now.....

3 divided by 7 equals 0 with remainder 3. From this result, we can write (0)(7) + 3 = 3

Cheers,
Brent
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Re: If x and y are different positive integers, which of the following COU  [#permalink]

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Top Contributor
1
GMATPrepNow wrote:
If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

i) When x is divided by y, the remainder is x
This occurs any time x < y
For example, if x = 5 and y = 7, then statement i becomes: When 5 is divided by 7, the remainder is 5
So true!

-----------------------------------------------------
ii) When 2x is divided by y, the remainder is x
Nice rule: If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

So, from statement ii, we can say: some possible values of 2x are: x, x + y, x + 2y, x + 3y, . . . etc
Let's examine the first option: 2x = x. Solve to get x = 0, but we're told x is POSITIVE No good.

Check the second option: 2x = x + y. Solve to get x = y. This means the remainder is y (aka x), but the remainder CANNOT be greater than the divisor. See the rule below:

When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

Check the third option: 2x = x + 2y. Solve to get x = 2y.
This means the remainder = 2y, which means the remainder is greater than the divisor (see rule above). No good.

In fact, we can see that, with all of the possible values of 2x, the remainder will be greater than the divisor.
So, statement ii is NOT true.

Scan the answer choices....eliminate B and E
-----------------------------------------------------
iii) When x+y is divided by x , the remainder is x-y
Some possible values of x+y are: (x-y), (x-y)+x, (x-y)+2x, (x-y)+3x, . . . etc
Let's examine the first option: x+y = x-y
Solve to get y = 0. No good.

Check the second option: x+y = (x-y)+x
Simplify: x+y = 2x - y
Solve to get: x = 2y

So, one possible case is: x = 6 and y = 3
Statement iii becomes: When (6 + 3) is divided by 6, the remainder is 3
So true!
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