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

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by Brent@GMATPrepNow » Thu Mar 07, 2019 10:15 am

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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

Difficulty level: 650 - 700
Source: www.gmatprepnow.com

Brent Hanneson - Creator of GMATPrepNow.com

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Source: Beat The GMAT — Data Sufficiency | Thu Mar 07, 2019 10:15 am

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

What is the value of Z?

by fskilnik@GMATH » Thu Mar 07, 2019 1:42 pm

Brent@GMATPrepNow wrote: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

Difficulty level: 650 - 700
Source: www.gmatprepnow.com

Beautiful problem, Brent. Congrats!

$$W,X,Y,Z\,\,\, \ge 1\,\,\,{\rm{different}}\,\,{\rm{ints}}\,\,\left( * \right)$$
$$\left\{ \matrix{
X = ZY + W\,\,,\,\,\,W < Y\,\,\,\left( {**} \right) \hfill \cr
\left( * \right)\,\,\,1 \le W < Y\,\,\, \Rightarrow \,\,\,Y \ge 2\,\,\,\left( {***} \right) \hfill \cr} \right.\,$$
$$? = Z$$
$$\left( 1 \right)\,\,X = W + 4\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,W + 4 = ZY + W\,\,\,\, \Rightarrow \,\,\,\,\,ZY = 4\,\,\,\,\mathop \Rightarrow \limits_{\left( {***} \right)}^{\left( * \right)} \,\,\,\,\,\,\left( {Y,Z} \right) = \left( {4,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
[The viability of the unique (Y,Z)=(4,1) is an examinerÂ´s burden. In this case it is nice to mention 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.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Mar 08, 2019 9:45 am

Brent@GMATPrepNow wrote: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

Difficulty level: 650 - 700
Source: www.gmatprepnow.com

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 X 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

Brent Hanneson - Creator of GMATPrepNow.com

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