Data Sufficiency

Is x odd?

(1) x/2 is not an integer

(2) 2x + 3 is odd.

The OA is C.

I don't have clear why that is the correct answer. Please, can any expert assist me with this DS question? Thanks.

Question: Is x odd?

Statement 1: x/2 is not an integer

Possibilities:

x is odd. For example, x = 3. 3/2 = 1.5, which is not an integer.

x is a fraction. For example, x = 1/2. (1/2)/2 = 1/4, which is not an integer.

Two different possibilities.

Insufficient.

(2) 2x + 3 is odd.

Possibilities:

x is even. For example, x = 2. 2(2) + 3 = 7, which is odd.

x is odd. For example, x = 1. 2(1) + 3 = 5, which is odd.

Two different possibilities.

Insufficient.

Statements Combined:

We have three possibilities from the two statements.

x is odd.

x is even.

x is a fraction.

If we can eliminate two, we will have sufficient information.

If we plug an even number into the expression from Statement 1 we get the following.

Even Number/2 = Integer

This is contrary to Statement 1. So, x cannot be even.

Statement 2 combines 2x + 3 to get an odd number. Since even + odd = odd, 2x must be even.

There is no way to multiply a fraction by 2 to get an even number. Consider the following examples.

2 * 1/2 = 1

2 * 1/3 = 2/3

2 * 1 1/2 = 3

So, x cannot be a fraction.

Therefore, x must be odd.

Sufficient.

The correct answer is C.