Which of these numbers can be expressed as a product of two or more prime numbers ?
(i) 15
(ii) 34568
(iii) `(15xx13)`

Options

A. only (ii)
B. only (iii)
C. only (i) and (ii)
D. all - (i), (ii) and (iii)

Answer

To determine which of the given numbers can be expressed as a product of two or more prime numbers, we will analyze each number step by step.

### Step 1: Analyze the number 15
1. **Factorization**: We can express 15 as the product of two prime numbers:
\[ 15 = 3 \times 5 \]
2. **Conclusion**: Since 15 can be expressed as a product of two prime numbers, it qualifies.

### Step 2: Analyze the number 34568
1. **Factorization**: We will perform prime factorization on 34568.
- First, we can divide by 2 (the smallest prime number):
\[ 34568 \div 2 = 17284 \]
- Continuing to divide by 2:
\[ 17284 \div 2 = 8642 \]
\[ 8642 \div 2 = 4321 \]
- Now, 4321 is not divisible by 2. We check for the next prime numbers:
- Dividing by 3, 5, 7, and finally we find:
\[ 4321 = 29 \times 149 \]
- Thus, the complete factorization is:
\[ 34568 = 2^3 \times 29 \times 149 \]
2. **Conclusion**: Since 34568 can be expressed as a product of three prime numbers (2, 29, and 149), it qualifies.

### Step 3: Analyze the expression (15xx13)
1. **Understanding the expression**: The expression `(15xx13)` indicates multiplication of 15 and 13.
2. **Factorization of 15**: As established earlier:
\[ 15 = 3 \times 5 \]
3. **Factorization of 13**: The number 13 is a prime number.
4. **Combining the factors**:
\[ 15 \times 13 = (3 \times 5) \times 13 = 3 \times 5 \times 13 \]
5. **Conclusion**: Since this expression can be expressed as a product of three prime numbers (3, 5, and 13), it qualifies.

### Final Conclusion
All three numbers (15, 34568, and 15xx13) can be expressed as a product of two or more prime numbers.

### Answer
All of the above numbers can be expressed as a product of two or more prime numbers.

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