If x, y and z are integers and xyz is divisible by 8, is x e

Question

If x, y and z are integers and xyz is divisible by 8, is x even?

(1) yz is divisible by 4.
(2) x,y and z are all NOT divisible by 4.

I am struggling to understand this question. The approach that I am trying to use is :

Statemenen 1 alone: INSUFFICIENT. Why? Because it doesn't say anything about x. x could be 1 and yz could be 4. x could be 8 and yz could e 4. So x can be ODD and EVEN both. Therefore, not sufficient. Is my understanding and concept correct for declining Statement 1?

Statement 2: If x,y and z are not divisible by 4, but they are divisible by 8 i.e they are divisible by at least 2^3. I am stuck after that. Can someone please help to tell me how to approach this mathematically?

The correct answer is B i.e statement 2 alone is sufficient.

fluke · Accepted Answer

Just think in terms of prime factors;

xyz is divisible by 8.
means; x*y*z must have at least 3 2's, doesn't matter from where those 2 come; the 3 2's are there.

Then think what possible ways can I get those three 2's.
x=8, y=1,z=1
We got 3 2's; this time from x.

x=1,y=8,z=1
I got 3 2's; this time from y.

x=1,y=1,z=8
I got 3 2's; this time from z.

There are many such combinations.
x=2,y=2,z=2
x=4,y=2,z=1
x=1,y=2,z=4

Q: Is x=even?

1. yz is divisible by 4.
This means that 
yz must have at least two 2's because 4 has two 2's.
Now,
if y=2, z=2; yz=4;
But from the stem we know xyz contains three 2's
Thus, the one additional 2 must come from x AND x becomes even. If "x" contains one 2, it becomes even.

But, we don't know whether y=2, z=2;
y may be 4 or 8 or 16
Then the odd/even for x or z doesn't make a difference. Because, y alone took care of both statements.

If y=8; y has 3 2's
Now even if x=1; z=1; xyz will be divisible by 8 AND yz will be divisible by 4.
Thus, we saw two different scenario where x may be an even or an odd.
Not sufficient.

2. x,y and z are all NOT divisible by 4.
This statement tells us that neither of x, y or z is divisible by 4. What does it mean?

It means; 
x does not contain two 2's. It may contain 0 2's or 1 2's.
y does not contain two 2's. It may contain 0 2's or 1 2's.
z does not contain two 2's. It may contain 0 2's or 1 2's.

However, we already know that xyz contain 3 2's.
Combining both the stem condition and statement 2, we can conclude that each variable has one AND only one 2 in its factors.

x=2
y=2
z=2

OR

x=2*3*7*11*13*13*13
y=2*5*43
z=2

The point is x will contain exactly one 2 in its factor. And if x contains one 2 in its factors, it must be even.
Sufficient.

Ans: "B"

Spidy001 · Answer

1. Not sufficient

yz is divisible by 4 = > yz is factor of 4

x may or may not be even.

x y z 
 2 4 1 => x is even
 3 4 2 => x is odd

2 . Sufficient

We need total of 3 2's in x,y and z. But we know each number is not divisible by 4.

=> each number has to have exactly one 2 .

Answer is B.

smartyman · Answer

Please help explain:
If x,y, and z are integers and xyz is divisible by 8, is x even?
1. yz is divisible by 4.
2. x,y, and z are all NOT divisible by 4.

I could not comprehend the explanation given by MGMAT for the second part of the statement. Please further explain. Thanks. Please ignore the difficulty tag.

blueseas · Answer

there is a typo in second statement. 
question is as follows:

If X, Y and z are integers and xyz is divisible by 8, isx even?

(1) yz is divisible by 4 
(2) X, Y, and z are all NOTdivisible b y 4

solution:

(1) yz is divisible by 4 
 If the value of yz = 4, then x should be even 
If the value of yz = 8, then x can be odd or even 
 Insufficient!

Quote:
 (2) X, Y, and z are all NOT divisible by 4 
 If x , y , z are multiples of 2, then xyz is divisible by 8 and x is even. This is the only condition which satisfies the conditions xyz divisible by 8 and x,y,z are all not divisible by 4  
 Sufficient!

hence B

Asifpirlo · Answer

In which book of Manhattans' did you find this problem can you tell me?
did you write the question correctly? check it once more..plz

smartyman · Answer

Sorry,

I misread the question; statement 2 should be
x,y, and z are all NOT divisible by 4.
then it makes sense.

Asifpirlo · Answer

thats best. sometimes it happens.

Somalika · Answer

Hi,

I am a bit confused with the second statement in the question.

The statement states - x,y and z are all NOT divisible by 4. I am confused because "all" is mentioned in the statement. 
Because of this I inferred that "Not all x, y,z are divisible by 4"- implying some can be divisible by 4. Hence I marked "E" as my answer.

Can anyone please clarify my doubt....

Thanks in advance.

stonecold · Answer

I marked the wrong choice
Lesson learnt => XYZ are not all divisible by 4 ≠ XYZ are all not divisible by 4
Damn..!!!

JeffTargetTestPrep · Answer

We are given that x, y, and z are integers and xyz is divisible by 8. We need to determine whether x is even, i.e., whether x is divisible by 2.

Statement One Alone:

yz is divisible by 4.

We don’t have enough information to determine whether x is even. For example, if yz = 4, then x could be 2, so that xyz = 8 is divisible by 8. However, if yz = 8, then x could be 3, so that xyz = 24 is divisible by 8. In the former case, x is even; in the latter case, x is odd. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x, y, z are all not divisible by 4.

Notice that 4 = 2^2 and 8 = 2^3. If x, y, and z are not divisible by 4, in order for the product, xyz, to be divisible by 8, each variable has to be divisible by 2. Thus, x must be even.

Answer: B

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.

If x, y and z are integers and xyz is divisible by 8, is x e

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

If x, y and z are integers and xyz is divisible by 8, is x e

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