coolmrityu wrote:Is the integer x even?

(1) x^2 is an even integer.
(2) x/4 is an odd integer.

Target question: Is the integer x even?

Statement 1: x^2 is an even integer.
If (x)(x) is even, then it must be the case that x is even

To demonstrate this, consider the following rules:
(odd)(odd) = odd
(odd)(even) = even
(even)(even) = even
So, if the product of two integers is even, then it must be the case that either both numbers are even or one is odd and one is even.
Since, x^2 requires us to multiply x by itself, we can rule out the possibility of one number being odd and the other number being even. This means that both numbers (x and x) are even.

Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x/4 is an odd integer.
We need only focus on the part that says, x/4 is an integer.
Let's say that x/4 = k (where k is an integer)
This means that x = 4k
In other words, x = (2)(2)k, which means x is a multiple of 2, which means x is definitely even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

coolmrityu wrote:Brent,

First of all, Thanks for prompt reply.

But i was facing problem with "Statement 1: x^2 is an even integer", which can take x as negative value. In that case one is unable to decide whether the integer x is even.

1. x2 = x(even)* x(even)
2. x2 = x(-ve)* x(-ve)

Please clarify.

Good question.
A lot of students feel that odd and even numbers must be positive. However, this is not the case.
Odd and even numbers can be positive, negative or neither (i.e., can equal 0).

Here are the odd #s: ...-5, -3, -1, 1, 3, 5, ...
Here are the even #s: ...-6, -4, -2, 0, 2, 4, 6, ...

So, as you can see, an integer can be both even and negative.

I hope that helps.

Cheers,
Brent

The formal definition says that,

Integer N is even if N = 2k for some integer k.

So, for example, -6 is even because we can rewrite -6 as 2(-3), where -3 is an integer.
Similarly, 0 is even because we can rewrite 0 as 2(0), where 0 is an integer.

Cheers,
Brent