Consider the following statements
I. If x and y are composite integers, so x + y is also composite.
II. If x and y are composite integers and `x gt y,` then x - y is also a composite integer.
III. If x and y are composite integers, so also is xy.
Of the above, the correct statement is/are

Options

A. Only III
B. I and II
C. All the three
D. None of these

Answer

To determine the correctness of the given statements regarding composite integers, let's analyze each statement step by step.

### Step 1: Understanding Composite Numbers
Composite numbers are integers greater than 1 that are not prime, meaning they have divisors other than 1 and themselves. Examples of composite numbers include 4, 6, 8, 9, 10, etc.

### Step 2: Analyzing Statement I
**Statement I:** If x and y are composite integers, then x + y is also composite.

**Counterexample:**
Let x = 4 (composite) and y = 6 (composite).
- x + y = 4 + 6 = 10, which is composite.
Now let’s try another example:
Let x = 4 and y = 4.
- x + y = 4 + 4 = 8, which is composite.
However, if we take x = 4 and y = 9,
- x + y = 4 + 9 = 13, which is prime.

Thus, Statement I is **not always true**.

### Step 3: Analyzing Statement II
**Statement II:** If x and y are composite integers and x > y, then x - y is also a composite integer.

**Counterexample:**
Let x = 9 (composite) and y = 4 (composite).
- x - y = 9 - 4 = 5, which is prime.

Thus, Statement II is **not always true**.

### Step 4: Analyzing Statement III
**Statement III:** If x and y are composite integers, then xy is also composite.

**Verification:**
Let x = 4 (composite) and y = 6 (composite).
- xy = 4 * 6 = 24, which is composite.
Let x = 9 and y = 4.
- xy = 9 * 4 = 36, which is composite.

Since the product of two composite numbers is always composite, Statement III is **always true**.

### Conclusion
- Statement I: False
- Statement II: False
- Statement III: True

The only correct statement is **Statement III**.