It is currently 20 Oct 2017, 11:32

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If x and y are positive integers, what is the remainder when

Author Message
TAGS:

### Hide Tags

Director
Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Joined: 26 Nov 2009
Posts: 956

Kudos [?]: 903 [3], given: 36

Location: Singapore
Concentration: General Management, Finance
Schools: Chicago Booth - Class of 2015
If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

04 Aug 2010, 15:17
3
KUDOS
13
This post was
BOOKMARKED
00:00

Difficulty:

85% (hard)

Question Stats:

52% (01:26) correct 48% (01:47) wrong based on 355 sessions

### HideShow timer Statistics

If x and y are positive integers, what is the remainder when x is divided by y?

(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4
[Reveal] Spoiler: OA

Kudos [?]: 903 [3], given: 36

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7676

Kudos [?]: 17369 [10], given: 232

Location: Pune, India
Re: If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

16 Nov 2011, 22:50
10
KUDOS
Expert's post
2
This post was
BOOKMARKED
nusmavrik wrote:
If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4

If you understand the concept of divisibility well, you can pretty much do this orally in less than 30 secs with a little bit of visualization. Divisibility involves grouping. Check these out first since I am explaining using the concept discussed in these posts:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/

Stmnt 1: When x is divided by 2y, the remainder is 4

When you divide x by 2y, you make groups with 2y balls in each and you have 4 balls leftover.
Instead, if you divide x by y, you may have 4 balls leftover or you may have fewer balls if y is less than or equal to 4 i.e. say if y = 3, you could make another group of 3 balls and you will have only 1 ball leftover. So you could have different remainders. Not sufficient.

Stmnt 2: When x + y is divided by y, the remainder is 4

When you make groups of y balls each from (x+y), the y balls make 1 group and you are left with x balls. If the remainder is 4, it means when you make groups of y balls each from x balls, you have 4 balls leftover.
Since the question asks us: how many balls are leftover when you make groups of y balls from x balls, you get your answer directly as '4'.
Sufficient.

_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 17369 [10], given: 232

Math Expert
Joined: 02 Sep 2009
Posts: 41892

Kudos [?]: 129003 [7], given: 12187

### Show Tags

04 Aug 2010, 22:46
7
KUDOS
Expert's post
6
This post was
BOOKMARKED
nusmavrik wrote:
If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4

Positive integer $$x$$ divided by positive integer $$y$$ yields remainder of $$r$$ can be expressed as $$x=yq+r$$. Question is $$r=?$$

(1) When x is divided by 2y, the remainder is 4. If $$x=20$$ and $$y=8$$ (satisfies the given statemnet as 20 divided by 2*8=16 yields reminder of 4), then $$x$$ divided by $$y$$ yields $$r=4$$ (20 divided by 8 yields remainder of 4) BUT if $$x=10$$ and $$y=3$$ (satisfies the given statemnet as 10 divided by 2*3=6 yields reminder of 4), then $$x$$ divided by $$y$$ yields $$r=1$$ (10 divided by 3 yields remainder of 1). Two different answers. Not sufficient.

(2) When x + y is divided by y, the remainder is 4 --> $$x+y=yp+4$$ --> $$x=y(p-1)+4$$ ($$x$$ is 4 more than multiple of $$y$$)--> this statement directly tells us that $$x$$ divided by $$y$$ yields remainder of 4. Sufficient.

Hope it's clear.
_________________

Kudos [?]: 129003 [7], given: 12187

Manager
Joined: 23 Oct 2011
Posts: 83

Kudos [?]: 114 [3], given: 34

### Show Tags

11 Nov 2011, 21:24
3
KUDOS
1
This post was
BOOKMARKED
Bunuel wrote:
That's a good question.
A. x=y(2k)+4, k any integer >=0.
B. x=y(p-1)+4, p any integer >=0.

Why A is not sufficient to determine the remainder and B is? Why did I use number plugging to show this in the first case and didn't in the second?

If we are told that x divided by y gives a remainder of 4, means x=yp+4 where p is integer >=0. We don't know x and y so p (quotient) can be any integer.

Look at equation A, the quotient is 2k, 2k is even. It can be rephrased as x divided by y will give the remainder of 4 IF quotient is even. But what about the cases when quotient is odd? We don't know that so we must check to determine this.

As for B. Quotient here is (p-1), which for integer values of p can give us ANY value: any even as well as any odd. So basically x=y(p-1)+4 is the same as x=yp+4. No need for double checking.

Hope it's clear.

Bunuel could we say the following? (Having in mind that the Remainder depends on the divisor)

(1) When x is divided by 2y, the remainder is 4

statement 1 ----> x=2*y*k+4, k integer

Therefore because the divisor has to be larger (not equal because it is stated that a reminder exists) than the remainder: 2*y>4 --> y>2 -->y>=3

If y (divisor) is smaller than 4 then the remainder changes and if it is larger than 4 the remainder is 4.

For example:
if y=3 then x=2*3*k+3+1, R=1
if y=4 then x=4*2*k+4+0, R=0
(if y=5 then x=2*5k+4, R=4)

Therefore Insufficient.

2) When x + y is divided by y, the remainder is 4

statement 2 ----> x+y=y*k+4, k integer

We are told that the remainder is 4, therefore y>=5! Which means that remainder will always be 4.

Kudos [?]: 114 [3], given: 34

Math Expert
Joined: 02 Sep 2009
Posts: 41892

Kudos [?]: 129003 [2], given: 12187

### Show Tags

24 Feb 2011, 12:40
2
KUDOS
Expert's post
1
This post was
BOOKMARKED
bugSniper wrote:
Bunuel wrote:
nusmavrik wrote:
If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4

Positive integer $$x$$ divided by positive integer $$y$$ yields remainder of $$r$$ can be expressed as $$x=yq+r$$. Question is $$r=?$$

(1) When x is divided by 2y, the remainder is 4. If $$x=20$$ and $$y=8$$ (satisfies the given statemnet as 20 divided by 2*8=16 yields reminder of 4), then $$x$$ divided by $$y$$ yields $$r=4$$ (20 divided by 8 yields remainder of 4) BUT if $$x=10$$ and $$y=3$$ (satisfies the given statemnet as 10 divided by 2*3=6 yields reminder of 4), then $$x$$ divided by $$y$$ yields $$r=1$$ (10 divided by 3 yields remainder of 1). Two different answers. Not sufficient.

(2) When x + y is divided by y, the remainder is 4 --> $$x+y=yp+4$$ --> $$x=y(p-1)+4$$ ($$x$$ is 4 more than multiple of $$y$$)--> this statement directly tells us that $$x$$ divided by $$y$$ yields remainder of 4. Sufficient.

Hope it's clear.

I'm confused. With the same proof given for the second point, could I not convince myself that 1 would suffice?
eg: x = yp + r
From (1) I could say x = 2yp + 4
(or) x = (2p)y + 4
Clearly this statement tells us that x is 4 more than a multiple of y as well. Why can I not convince myself at this step that the statement would suffice? Although, the statement does not suffice since the logic fails when you plug in (x=10,y=3) .

That's a good question.
A. x=y(2k)+4, k any integer >=0.
B. x=y(p-1)+4, p any integer >=0.

Why A is not sufficient to determine the remainder and B is? Why did I use number plugging to show this in the first case and didn't in the second?

If we are told that x divided by y gives a remainder of 4, means x=yp+4 where p is integer >=0. We don't know x and y so p (quotient) can be any integer.

Look at equation A, the quotient is 2k, 2k is even. It can be rephrased as x divided by y will give the remainder of 4 IF quotient is even. But what about the cases when quotient is odd? We don't know that so we must check to determine this.

As for B. Quotient here is (p-1), which for integer values of p can give us ANY value: any even as well as any odd. So basically x=y(p-1)+4 is the same as x=yp+4. No need for double checking.

Hope it's clear.
_________________

Kudos [?]: 129003 [2], given: 12187

Senior Manager
Joined: 25 Feb 2010
Posts: 450

Kudos [?]: 110 [1], given: 10

### Show Tags

04 Aug 2010, 19:53
1
KUDOS
1. IMO B

the second one:
(x+y)/y= b+4,
x/y+y/y=b+4,
x/y+1=b+4
or x/y=b+3. the remainder is 3
with this one we agree, it is sufficient.
_________________

GGG (Gym / GMAT / Girl) -- Be Serious

Its your duty to post OA afterwards; some one must be waiting for that...

Kudos [?]: 110 [1], given: 10

Math Forum Moderator
Joined: 20 Dec 2010
Posts: 1964

Kudos [?]: 2051 [1], given: 376

### Show Tags

24 Feb 2011, 12:53
1
KUDOS
B is sufficient by using the rule of remainders additive property:
(x+y)/y leaves a remainder of 4.
Means: remainder left by x/y + remainder left by y/y = 4
remainder left by x/y+0=4
remainder left by x/y=4

At least B is Sufficient.
_________________

Kudos [?]: 2051 [1], given: 376

Intern
Joined: 24 Jul 2007
Posts: 5

Kudos [?]: [0], given: 5

### Show Tags

04 Aug 2010, 20:48
isnt st 2 enugh?
(X+y)=y(p)+4
so, x= y(p-1)+4; so means remainder of x/y is 4

Please correct me if am wrng. For st1, I cant find an x/y using x=2y(m)+4

Kudos [?]: [0], given: 5

Director
Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Joined: 26 Nov 2009
Posts: 956

Kudos [?]: 903 [0], given: 36

Location: Singapore
Concentration: General Management, Finance
Schools: Chicago Booth - Class of 2015

### Show Tags

05 Aug 2010, 00:14
trapped !

Its the quotient that decreases (minus 1). The remainder is unaffected.

E.g (3 + 5) /5 ----> remainder = 3, quotient = 1
3/5 ------> remainder = 3, quotient = 0

onedayill wrote:
1. IMO B

the second one:
(x+y)/y= b+4,
x/y+y/y=b+4,
x/y+1=b+4
or x/y=b+3. the remainder is 3
with this one we agree, it is sufficient.

Kudos [?]: 903 [0], given: 36

Intern
Joined: 23 Jan 2011
Posts: 8

Kudos [?]: 2 [0], given: 3

### Show Tags

24 Feb 2011, 12:30
Bunuel wrote:
nusmavrik wrote:
If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4

Positive integer $$x$$ divided by positive integer $$y$$ yields remainder of $$r$$ can be expressed as $$x=yq+r$$. Question is $$r=?$$

(1) When x is divided by 2y, the remainder is 4. If $$x=20$$ and $$y=8$$ (satisfies the given statemnet as 20 divided by 2*8=16 yields reminder of 4), then $$x$$ divided by $$y$$ yields $$r=4$$ (20 divided by 8 yields remainder of 4) BUT if $$x=10$$ and $$y=3$$ (satisfies the given statemnet as 10 divided by 2*3=6 yields reminder of 4), then $$x$$ divided by $$y$$ yields $$r=1$$ (10 divided by 3 yields remainder of 1). Two different answers. Not sufficient.

(2) When x + y is divided by y, the remainder is 4 --> $$x+y=yp+4$$ --> $$x=y(p-1)+4$$ ($$x$$ is 4 more than multiple of $$y$$)--> this statement directly tells us that $$x$$ divided by $$y$$ yields remainder of 4. Sufficient.

Hope it's clear.

I'm confused. With the same proof given for the second point, could I not convince myself that 1 would suffice?
eg: x = yp + r
From (1) I could say x = 2yp + 4
(or) x = (2p)y + 4
Clearly this statement tells us that x is 4 more than a multiple of y as well. Why can I not convince myself at this step that the statement would suffice? Although, the statement does not suffice since the logic fails when you plug in (x=10,y=3) .

Kudos [?]: 2 [0], given: 3

Manager
Joined: 09 Nov 2011
Posts: 127

Kudos [?]: 65 [0], given: 16

Re: If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

17 Nov 2011, 09:23
Karishma...Really impressive reply..!! Very precise..and easily understandable...went through your post too...had never imagined division from this perspective...i think this is a better approach to these questions..! Thanks for sharing!
_________________

Time to play the game...

Kudos [?]: 65 [0], given: 16

Intern
Status: Stay Hungry, Stay Foolish.
Joined: 05 Sep 2011
Posts: 40

Kudos [?]: 9 [0], given: 6

Location: India
Concentration: Marketing, Social Entrepreneurship
Re: If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

17 Nov 2011, 09:58
I must admit,initially,i did get trapped in option A & B both and would've answered both are sufficient,but carefully after evaluating,realized, only B suffices.
A doesn't.

Kudos [?]: 9 [0], given: 6

Senior Manager
Joined: 12 Oct 2011
Posts: 256

Kudos [?]: 61 [0], given: 110

Re: If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

17 Dec 2011, 19:53
Yes a tricky question. Karishma, thank you for the detailed explanation. The concept of "grouping" applied to division, although new to me, is easily understandable and very simple indeed.
_________________

Consider KUDOS if you feel the effort's worth it

Kudos [?]: 61 [0], given: 110

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 746

Kudos [?]: 2083 [0], given: 123

Re: If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

29 May 2015, 02:32
Quote:
If x and y are positive integers, what is the remainder when x is divided by y?

(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4

Here's an alternate solution to this question, without using number-plugging:

The question asks us the remainder for x/y

This means, in the expression x = ky + r . . . (1) (where k is the quotient and r is the remainder)

We have to find the value of r, where 0 =<r < y

Analyzing St. 1
When x is divided by 2y, the remainder is 4

That is, x = (m)(2y) + 4 . . . (2)

Now, upon comparing Equations (1) and (2), can we say r = 4? Let's see:

We know that remainder is always less than the divisor.

Since r is the remainder in Equation 1, we can be sure that r < y

Similarly, since 4 is the remainder in Equation 2, we can be sure that 4 < 2y. This gives us, y > 2

So, y can be 3, 4, 5, 6, 7 etc.

Now, if y = 3, then r must be less than 3. So, upon comparing (1) and (2), you'll not say that r = 4. Instead, you'll break down 4 into 3 + 1. So, if y = 3, remainder r = 1

If y = 4, then r must be less than 4. So, upon comparing (1) and (2), you'll not say that r = 4. Instead, you'll write 4 as 4 + 0. So, if y = 4, remainder r = 0

If y > 4, then 4 becomes a valid value for remainder r. So, in this case, upon comparing (1) and (2), we can say that r = 4.

Thus, we see that r can be 0, 1 or 4. Since we have not been able to determine a unique value of r, St. 1 is not sufficient.

Analyzing St. 2
When x + y is divided by y, the remainder is 4

That is, x + y = qy + 4
Or, x = (q - 1)y + 4 . . . (3)

Equation (3) conveys that when x is divided by y, the remainder is 4. So, St. 2 is sufficient to find the value of the remainder.

Please note that the way we processed St. 1 was different from the way we processed St. 2, because in St. 1 the divisor was 2y, whereas in St. 2, the divisor was y (the same divisor as in the question)

Hope this alternate solution was useful!

Best Regards

Japinder
_________________

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Kudos [?]: 2083 [0], given: 123

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16635

Kudos [?]: 273 [0], given: 0

Re: If x and y are positive integers, what is the remainder when [#permalink]

### Show Tags

03 Oct 2017, 06:38
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).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 273 [0], given: 0

Re: If x and y are positive integers, what is the remainder when   [#permalink] 03 Oct 2017, 06:38
Display posts from previous: Sort by