number properties problem

[theboyleman32]

Any help on this problem is greatly appreciated!

[ceilidh.erickson]

If the product xyz is even, one of them, two of them, or all three of them could be even. If we want to know if z is even, we can test numbers, or think conceptually.

1) x/y = z

Think conceptually:
We can rearrange this to say x = yz. If x is equal to the product of y and z, then xyz = x^2. Since we know that xyz is even, it must also follow that x^2 is even, and thus x is even. (An even squared = even; odd squared = odd).

Since x is even, and it's the product of y and z, it must follow that either y or z is even, or both (because odd*odd = odd, so they can't both be odd). It's possible, though, that y is even and z is odd.

Test numbers:
x=4, y=2, z=2 --> 4/2 = 2
yes, z is even

x=6, y=2, z=3 --> 6/2 = 3
no, z is no even

Insufficient.



2) z = xy

Think conceptually:
If z is equal to the product of x and y, then xyz = z^2. Since we know that xyz is even, it must also follow that z^2 is even, and thus z is even. (An even squared = even; odd squared = odd).

Or thinking about it another way, if x and y were both odd, then z would be odd, and the product xyz would be odd, which can't be true. Thus, either x or y or both must be even, making z even.

Test numbers:
x=2, y=3, z=6 --> 6 = 2*3

There are no numbers that we could test here that would give us an even product of xyz, but not an even product of xy.

Sufficient.

The answer is B.

[Brent@GMATPrepNow]

The product of integers x, y and z is even. Is z even?

1) x/y = z
2) z = xy

Target question: Is z even?

Given: The product of integers x, y and z is even.
KEY CONCEPT: If xyz is even, then it MUST be the case that at least one of x,y and z is even.. Otherwise, the product xyz would be odd

Statement 1: x/y = z
There are several values of x, y and z that satisfy this condition. Here are two:
Case a: x= 8, y = 2 and z = 3 (notice that xyz = 48, which is even). In this case z IS even
Case b: x= 6, y = 2 and z = 3 (notice that xyz = 36, which is even). In this case z is NOT even
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: z = xy
We've already concluded that at least one of x,y and z is even, so either x is even, y is even, or z is even.
Let's consider all 3 cases:
- if x is even, and if z = xy, then z = (some even number)(y), which means z is even
- if y is even, and if z = xy, then z = (x)(some even number)), which means z is even
- if z is even, then z is even
In all 3 possible cases, z is even, so IT MUST BE THE CASE THAT z is even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

[Matt@VeritasPrep]

Another approach!

Start with the stem, which tells you that at least one of x, y, and z is even.

S1:: Rearrange this to x = y * z. We know that AT LEAST one of the integers is even. If x is even, then y * z is even (since x = y * z). If y * z is even, then either y or z is even, but we don't know which one. INSUFFICIENT

S2:: As before, we already know that at least one of the three integers is even. If it's z, we're set. Now suppose that it isn't, which means that either x or y is even. But z = x * y, so z = even * x or z = even * y, so z is also even. So in either case, z is even. SUFFICIENT!