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

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

by melguy » Tue Nov 01, 2016 4:07 am
Please help with question below

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,z are all not divisible by 4

Statement 1

We do not know anything about x
Not Sufficient

Statement 2
I am a bit confused with the language of the question. Is it suggesting that only possible value of x y z is restricted to 2 each? Hence the statement is sufficient?

But possible values of x y z can be

x=6 y=2 z=2
x=3 y=6 z=2

So why is the answer B
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by fiza gupta » Tue Nov 01, 2016 5:58 am
(2) second statement implies that not all x,y,z are divisible by 4
to make xyz divisible by 8 all hhas to be multiple of 2.
SUFFICIENT
Last edited by fiza gupta on Tue Nov 01, 2016 9:53 am, edited 1 time in total.
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by [email protected] » Tue Nov 01, 2016 9:42 am
Hi melguy,

The wording in Fact 2 is a bit clunky, but if the intent of this question is that the correct answer is supposed to be B, then here is how you're meant to interpret Fact 2:

To start, we know that X, Y and Z are all INTEGERS and that (X)(Y)(Z) is divisible by 8. This means that when we prime-factor (X)(Y)(Z) we'll have AT LEAST three 2s. We're asked if X is EVEN. This is a YES/NO question.

1) (Y)(Z) is divisible by 4.

IF...
X=2, Y=2, Z=2, then the answer to the question is YES.
X=1, Y=4, Z=4, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

2) X, Y and Z are all NOT divisible by 4

Since this Fact states that the variables are "all not divisible by 4", then whatever these three variables are... none of them is divisible by 4. Thus, none is a multiple of 4, although they could all be even. Remember though - we're told that (X)(Y)(Z) is divisible by 8, so there has to be three 2s in the prime-factorization of that product. If none of the variables is divisible by 4, then each variable could contain no more than one 2 in its prime-factorization. With three variables and three 2s needed, each variable MUST contain one 2 - which means that each variable MUST be EVEN. The answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

Final Answer: B

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by Jay@ManhattanReview » Thu Jan 05, 2017 11:05 pm
melguy wrote:Please help with question below

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,z are all not divisible by 4

Statement 1

We do not know anything about x
Not Sufficient

Statement 2
I am a bit confused with the language of the question. Is it suggesting that only possible value of x y z is restricted to 2 each? Hence the statement is sufficient?

But possible values of x y z can be

x=6 y=2 z=2
x=3 y=6 z=2

So why is the answer B
Hi melguy,

As far as the question "If x, y and z are integers and xyz is divisible by 8, is x even?" is concerned, it is clear that it does not suggest that each of x, y, and z are 2. All it implies: (1) x, y and z are integers. (2) [x.y.z] is a multiple of 8.

Till we see a statement, we have to only think that at least one among x, y, and z is even. x can be even or odd.

Let us see each statement one by one.

S1: Given that yz is divisible by 4.

If x is a multiple of 2 (even), xyz is a multiple of 8, thus the answer is YES.

However, if yz itself is a multiple of 8, x may or may not be even; it may be odd. The answer is NO. Insufficient.

S2: Given that x, y, and z are all not divisible by 4, we have to think of how will the product of x, y, and z would be a multiple of 8 = 2^3?

We can deduce that each of x, y, and x must be a multiple of 2, so that x.y.z would be a multiple of 8 = 2^3. Thus, x is even. Sufficient.

Hope this helps!

-Jay

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by Brent@GMATPrepNow » Sun Jan 08, 2017 3:01 pm
melguy wrote:Please help with question below

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,z are all not divisible by 4

Statement 1

We do not know anything about x
Not Sufficient

Statement 2
I am a bit confused with the language of the question. Is it suggesting that only possible value of x y z is restricted to 2 each? Hence the statement is sufficient?

But possible values of x y z can be

x=6 y=2 z=2
x=3 y=6 z=2

So why is the answer B
Be careful. Your second example (x=3 y=6 z=2) does not meet the given condition that xyz is divisible by 8.
Here, xyz = (3)(6)(2) = 36, and 36 is not divisible by 8
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by Jeff@TargetTestPrep » Wed Jan 11, 2017 7:12 am
melguy wrote:Please help with question below

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,z are all not divisible by 4
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

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