Which of the following is INCORRECT?

Options

A. If x is a rational number, such that the prime factorisation of denominator is not in the form `2^(n) 5^(m)`, (where m and n are non-negative integers), then it has a decimal expansion which is non-terminating and repeating.
B. `5 + sqrt(2)` is an irrational number.
C. Every composite number can be expressed as a product of primes.
D. None of these

Answer

To determine which of the statements is incorrect, let's analyze each option step by step.

### Step 1: Analyze Option 1
**Statement:** If x is a rational number such that the prime factorization of the denominator is not in the form of \(2^m \times 5^n\) (where \(m\) and \(n\) are non-negative integers), then it has a decimal expansion which is non-terminating and repeating.

**Analysis:**
- A rational number has a terminating decimal expansion if its denominator (in simplest form) can be expressed as \(2^m \times 5^n\).
- If the denominator cannot be expressed in this form, the decimal expansion will be non-terminating and repeating.
- Therefore, this statement is **correct**.

### Step 2: Analyze Option 2
**Statement:** \(5 + 5 + \sqrt{2}\)

**Analysis:**
- The expression simplifies to \(10 + \sqrt{2}\).
- Since \(\sqrt{2}\) is an irrational number, adding it to a rational number (10) results in an irrational number.
- Therefore, this statement is **correct**.

### Step 3: Analyze Option 3
**Statement:** Every composite number can be expressed as a product of primes.

**Analysis:**
- A composite number is defined as a positive integer that has at least one positive divisor other than one or itself.
- According to the Fundamental Theorem of Arithmetic, every integer greater than 1 can be expressed as a product of prime numbers.
- Therefore, this statement is **correct**.

### Step 4: Analyze Option 4
**Statement:** None of the above statements is incorrect.

**Analysis:**
- Since all the previous statements (1, 2, and 3) are correct, this statement is also correct.

### Conclusion
Since all the statements provided in options 1, 2, and 3 are correct, the answer to the question "Which of the following is INCORRECT?" is option 4: **None of these.**

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