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Manager  Joined: 16 Jul 2009
Posts: 176
If w, x, y, and z are non-negative integers, each less than  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 70% (02:27) correct 30% (02:36) wrong based on 755 sessions

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If w, x, y, and z are non-negative integers, each less than 3, and $$w(3^3) + x(3^2) + y(3) + z = 34$$, then w+z=

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: Number Properties  [#permalink]

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abhi758 wrote:
If w, x, y, and z are non-negative integers, each less than 3, and $$w(3^3) + x(3^2) + y(3) + z = 34$$, then w+z=

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$w(3^3) + x(3^2) + y(3)$$ in any case is a multiple of 3, so we have: {multiple of 3} + z = 34, as given that $$0\leq{z}<3$$, then $$z=1$$.

Next, $$w$$ cannot be 2 or 0 as for $$w=2$$ the sum is more than 34 and for $$w=0$$ others cannot total to 34. Hence $$w=1$$.

$$w+z=2$$.

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Intern  Joined: 26 Jan 2014
Posts: 40
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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I try a little bit first.

27W+9X+3Y+Z=34

1. If W\neq{0}m, then if X=2,Y=2, Z=2 the answer is smaller than 34, which means this hypothesis didn't exist.
2. If W=1, then I try to put some number in the formula. (1) X=1 ,no, already exceed 34. (2）X=0 Y=2 Z=1 BINGO

Thus: W+Z=2
Intern  Joined: 19 Aug 2013
Posts: 14
Schools: Kellogg PT '19
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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I agree with z=1, what if w=0, x=3 and y=2?
Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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udaymeka36 wrote:
I agree with z=1, what if w=0, x=3 and y=2?

We are told that w, x, y, and z are non-negative integers, each less than 3.
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Concentration: General Management, Technology
GMAT 1: 630 Q45 V32 WE: Information Technology (Computer Software)
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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Hi Bunuel,

Non negative includes 0 as well right...
w=0,x=3,y=2,z=1 works fine
Intern  Joined: 12 Mar 2015
Posts: 2
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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1
Hi Janani,

The question says that each of the variables (w,x,y,z) should be less than 3.

The assumption you have taken is x = 3, which goes against the rules of the question.

Hence, C is the answer.
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Joined: 13 Jun 2012
Posts: 199
Location: United States
WE: Supply Chain Management (Computer Hardware)
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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Bunuel wrote:
abhi758 wrote:
If w, x, y, and z are non-negative integers, each less than 3, and $$w(3^3) + x(3^2) + y(3) + z = 34$$, then w+z=

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$w(3^3) + x(3^2) + y(3)$$ in any case is a multiple of 3, so we have: {multiple of 3} + z = 34, as given that $$0\leq{z}<3$$, then $$z=1$$.

Next, $$w$$ cannot be 2 or 0 as for $$w=2$$ the sum is more than 34 and for $$w=0$$ others cannot total to 34. Hence $$w=1$$.

$$w+z=2$$.

Hello Bunuel ! I am just little confused here. Why z cannot be 2? {multiple of 3} + z = 34 this means z cannot be 3.
Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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1
1
Turkish wrote:
Bunuel wrote:
abhi758 wrote:
If w, x, y, and z are non-negative integers, each less than 3, and $$w(3^3) + x(3^2) + y(3) + z = 34$$, then w+z=

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$w(3^3) + x(3^2) + y(3)$$ in any case is a multiple of 3, so we have: {multiple of 3} + z = 34, as given that $$0\leq{z}<3$$, then $$z=1$$.

Next, $$w$$ cannot be 2 or 0 as for $$w=2$$ the sum is more than 34 and for $$w=0$$ others cannot total to 34. Hence $$w=1$$.

$$w+z=2$$.

Hello Bunuel ! I am just little confused here. Why z cannot be 2? {multiple of 3} + z = 34 this means z cannot be 3.

34 is 1 more than a multiple of 3, if z = 2, then {multiple of 3} + 2 is 2 more than a multiple of 3.
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Intern  Joined: 01 Feb 2016
Posts: 9
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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Non negative means that digits can be zero (zero is neither positive, nor negative, thus it is non-negative)

I am getting 1. Here is my logic (extending previous one)

w(3^3)+x(3^2)+y(3)+z=34
27w+9x+3y+z=34
3(9w+3x+y)+z=34. Thus (multiple of 3)+z=34. or 3xP+z=34. As z has to be between 0 and less than 3, it has to be equal to 1 because If P is 11, then 33+1=34. If P is less than 11 (say 10) then z has to be 4 (3*10+4=34, which is against the condition stated in problem that z has to be between 0 and less than 3)

Going to above equation, 3*11+1=4 => 9w+3x+y=11.
Repeating the previous logic, 3(3w+x)+y=11. Thus y must be 2. (3.3+2=11)

Thus 3w+x=2. Thus w must be 0 else any value of w > 0 would not make equation equal to 2 even if x=0.

So z+w=1 (ans A)

Does any one disagree?
Intern  Joined: 03 Jun 2016
Posts: 10
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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Bunuel wrote:
udaymeka36 wrote:
I agree with z=1, what if w=0, x=3 and y=2?

We are told that w, x, y, and z are non-negative integers, each less than 3.

if w=z=1, then why 2 different variables are given if they have same value ?
Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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mahawarchirag wrote:
Bunuel wrote:
udaymeka36 wrote:
I agree with z=1, what if w=0, x=3 and y=2?

We are told that w, x, y, and z are non-negative integers, each less than 3.

if w=z=1, then why 2 different variables are given if they have same value ?

Unless otherwise specified, different variables can represent the same number.
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Intern  Joined: 03 Jun 2016
Posts: 10
Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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We are told that w, x, y, and z are non-negative integers, each less than 3.[/quote]

if w=z=1, then why 2 different variables are given if they have same value ?[/quote]

Unless otherwise specified, different variables can represent the same number.[/quote]

Have encountered such situation for the 1st time. Thanks for sharing .
Senior Manager  S
Joined: 24 Jun 2016
Posts: 344
GMAT 1: 770 Q60 V60 GPA: 4
If w, x, y, and z are non-negative integers, each less than  [#permalink]

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Since 34/3 remainder 1, z/3 remainder 1. And since z<3, z=1
So 27w + 9x + 3y = 33, or 9w + 3x + y = 11
Since x<=2 and y<=2, 3x+y<=8, so 9w >0 0. Since 9w<=11, 9w=9, so w is 1. Therefore $$w+z = 1+ 1 = 2$$

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Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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abhi758 wrote:
If w, x, y, and z are non-negative integers, each less than 3, and $$w(3^3) + x(3^2) + y(3) + z = 34$$, then w+z=

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

We can simplify our given equation, and we have:

27w + 9x + 3y + z = 34

We are given that w, x, y, and z are non-negative integers less than 3; thus they can be only 0, 1, or 2. We see that if w = 2, then 27w = 54 and that would be too big for the sum to be 34.

If w = 1, then 27w = 27, and in order to satisfy the equation, 9x + 3y + z must be 7.

If that is the case, then x must be 0. Thus, 3y + z = 7.

At this point we see that y must be 2 and z must be 1 in order to satisfy the equation.

Thus, w + z = 1 + 1 = 2.

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Re: If w, x, y, and z are non-negative integers, each less than  [#permalink]

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