Last visit was: 19 Nov 2025, 13:48 It is currently 19 Nov 2025, 13:48
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
Close
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
Your Progress

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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
805+ Level|   Remainders|      
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 31 Oct 2025
Posts: 6,739
Own Kudos:
35,353
 [25]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,739
Kudos: 35,353
 [25]
W, X, Y and Z are four different positive integers. 06 Mar 2019, 11:28
4
Kudos
Add Kudos
20
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

30% (02:40) correct 70% (02:43) wrong based on 194 sessions

History

Date
Time
Result
Not Attempted Yet
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W.
What is the value of Z?

1) W = X – 4
2) W + Z = 4
ShowHide Answer
Official Answer
A
Most Helpful Reply
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 15 Nov 2025
Posts: 11,238
Own Kudos:
43,706
 [9]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,238
Kudos: 43,706
 [9]
W, X, Y and Z are four different positive integers. 06 Mar 2019, 20:14
4
Kudos
Add Kudos
5
Bookmarks
Bookmark this Post
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W.
What is the value of Z?
So, when we put this information in an equation form, we get X=YZ+W

1) W = X – 4
Now X=YZ+W, substitute W=X-4 in this..
So X=YZ+X-4.....YZ=4
We know that Y and Z are different positive integers, so 4 can be only 1*4
Thus Y and Z can be 1 or 4... But if Y, that is the divider, is 1, then there will be no remainder as all numbers are multiple of 1.
Thus Y is 4 and Z is 1..
Sufficient

2) W + Z = 4
Substitute this in X=YZ+W....X=YZ+4-Z
INSUFFICIENT

A..
just for information.. the values from 2 statements will become X=YZ+W.....
7=1*4+3
General Discussion
User avatar
Archit3110
User avatar
Major Poster
Joined: 18 Aug 2017
Last visit: 19 Nov 2025
Posts: 8,422
Own Kudos:
4,982
 [1]
Given Kudos: 243
Status:You learn more from failure than from success.
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1: 545 Q79 V79 DI73
GMAT Focus 2: 645 Q83 V82 DI81
GPA: 4
WE:Marketing (Energy)
GMAT Focus 2: 645 Q83 V82 DI81
Posts: 8,422
Kudos: 4,982
 [1]
W, X, Y and Z are four different positive integers. 07 Mar 2019, 02:35
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GMATPrepNow
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W.
What is the value of Z?

1) W = X – 4
2) W + Z = 4
Show more

#1
given info we can say
X=YZ+W
w=x-4
so
yz=4
sufficient
z=1 and y=4

#2
w+z=4
x=4y-yw+w
not sufficient
IMO A
User avatar
fskilnik
User avatar
Joined: 12 Oct 2010
Last visit: 03 Jan 2025
Posts: 885
Own Kudos:
1,801
Given Kudos: 57
Status:GMATH founder
Expert
Expert reply
Posts: 885
Kudos: 1,801
W, X, Y and Z are four different positive integers. 07 Mar 2019, 14:57
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GMATPrepNow
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W.
What is the value of Z?

1) W = X – 4
2) W + Z = 4
Show more
Beautiful problem, Brent. Kudos!


\(W,X,Y,Z\,\,\, \ge 1\,\,\,{\rm{different}}\,\,{\rm{ints}}\,\,\left( * \right)\)

\(\left\{ \matrix{\\
X = ZY + W\,\,,\,\,\,W < Y\,\,\,\left( {**} \right) \hfill \cr \\
\left( * \right)\,\,\,1 \le W < Y\,\,\, \Rightarrow \,\,\,Y \ge 2\,\,\,\left( {***} \right) \hfill \cr} \right.\,\)

\(? = Z\)


\(\left( 1 \right)\,\,X = W + 4\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,W + 4 = ZY + W\,\,\,\, \Rightarrow \,\,\,\,\,ZY = 4\,\,\,\,\mathop \Rightarrow \limits_{\left( {***} \right)}^{\left( * \right)} \,\,\,\,\,\,\left( {Y,Z} \right) = \left( {4,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)

(The viability of the unique (Y,Z)=(4,1) is an examiner´s burden. In this case it is nice to mention, as cheetan2u did, that (X,Y,Z,W)=(7,4,1,3) is viable. The problem is perfect!)


\(\left( 2 \right)\,\,W + Z = 4\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {X,Y,Z,W} \right) = \left( {7,2,3,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 3\,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {X,Y,Z,W} \right) = \left( {7,4,1,3} \right)\,\,\,\, \Rightarrow \,\,\,? = 1 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{INSUFF}}{\rm{.}}\)


The correct answer is (A).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 31 Oct 2025
Posts: 6,739
Own Kudos:
35,353
 [1]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,739
Kudos: 35,353
 [1]
W, X, Y and Z are four different positive integers. Updated on: 04 Sep 2022, 12:21
Originally posted by BrentGMATPrepNow on 08 Mar 2019, 10:58.
Last edited by BrentGMATPrepNow on 04 Sep 2022, 12:21, edited 2 times in total.
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
GMATPrepNow
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W.
What is the value of Z?

1) W = X – 4
2) W + Z = 4
Show more

Target question: What is the value of Z?

Given: When X is divided by Y, the quotient is Z and the remainder is W.
------ASIDE-------
There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
---------------------------------------
So, from the given information, we can write: X = YZ + W

Statement 1: W = X – 4
Take X = YZ + W and replace W with X - 4 to get: X = YZ + X - 4
Subtract X from both sides: 0 = YZ - 4
Rewrite as: 4 = YZ

We're told that Y and Z are DIFFERENT positive INTEGERS. So, there are only 2 possible cases:
case i: Y = 1 and Z = 4
case ii: Y = 4 and Z = 1

case i yields a CONTRADICTION.
If Y = 1, then we are dividing X by 1, and if we divide by 1, the remainder will always be ZERO.
In other words, if Y = 1, then W = 0
However, we are told that W is a POSITIVE integer.
So, we can definitely rule out case i, which means it MUST be the case that Y = 4 and Z = 1 (case ii)
So, the answer to the target question is Z = 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: W + Z = 4
Since W and Z are DIFFERENT positive INTEGERS. So, there are only 2 possible cases:
case i: W = 1 and Z = 3
case ii: W = 3 and Z = 1
Let's check each case to see whether each case yields a contradiction.

case i: W = 1 and Z = 3
We get: "When X is divided by Y, the quotient is 3 and the remainder is 1"
So, for example, it could be the case that X = 7 and Y = 2
We get: When 7 is divided by 2, the quotient is 3 and the remainder is 1
So, case i is possible. (In this case, the answer to the target question is Z = 3

case ii: W = 3 and Z = 1
We get: "When X is divided by Y, the quotient is 1 and the remainder is 3"
So, for example, it could be the case that X = 7 and Y = 4
We get: When 7 is divided by 4, the quotient is 1 and the remainder is 3
So, case ii is also possible. (In this case, the answer to the target question is Z = 1

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

RELATED VIDEO
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,587
Own Kudos:
1,079
Posts: 38,587
Kudos: 1,079
W, X, Y and Z are four different positive integers. 17 Jul 2025, 21:59
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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.
Moderators:
Bunuel
Math Expert
105390 posts
kingbucky
496 posts