W, X, Y and Z are four different positive integers.

Question

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

chetan2u · Accepted Answer

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

Archit3110 · Answer

#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

fskilnik · Answer

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.

BrentGMATPrepNow · Answer

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 
 https://www.youtube.com/watch?v=XCxgIR042tE

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

W, X, Y and Z are four different positive integers.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

W, X, Y and Z are four different positive integers.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards