Data Sufficiency

[email protected] wrote:The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?

(1) xz is even

(2) y is even.


Is testing numbers the ideal way for solving this question?

Answer-D

Testing numbers can take a bit of time, especially when there are 3 variables.

Here's a more formal approach.

Target question: Is z even?

Given: x is a factor of y, and y is a factor of z.

There's a nice rule that says, "If D is a factor (divisor) of N, then N = kD for some integer k"
So, if x is a factor of y, then y = kx for some integer k.
Also, if y is a factor of z, then z = jy for some integer j

Statement 1: xz is even
This sets up two possible cases (x is even or z is even). We'll examine both:
case a: x is even.
If x is even, then kx is even, which means y is even (since y=kx).
If y is even, then jy is even, which means z is even (since z=jy).
case b: z is even
Since both possible cases yield the same answer to the target question, statement 1 is SUFFICIENT

Statement 2: y is even
If y is even, then jy is even, which means z is even (since z=jy).
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Hi shibsriz,

This DS question hides a great Number Property that's worth knowing. Brett's solution is absolutely correct and explains all the details behind the definition of factors.

Here's the Number Property:

Even integers DO NOT divide evenly into odd integers. In terms of factors, even integers ARE NOT factors of odd integers.
For example, 2, 4, 6, etc. DO NOT divide into 7, 11, 27, 83, etc.
You can use this rule against the Facts in this DQ question.

We're told that x is a factor of y and y is a factor of z. We're asked if z is even. This is a Yes/No question.

Fact 1: xz = even

This mean either we have "one even, one odd" or "two evens"

If x = even, then y = even and z = even. This is because even numbers don't divide into odd numbers. The answer to the question is YES.
If x = odd, then z = even. The answer to the question is Yes.
Fact 1 is SUFFICIENT

Fact 2: y = even

Since y = even, then z = even. This is because that even numbers don't divide into odd numbers. The answer to the question is YES.
Fact 2 is SUFFICIENT

Final Answer: D

GMAT assassins aren't born, they're made,
Rich