If P is a prime number and P divides ab i.e., `P|(ab)` where a and b are integers, then

Options

A. `P|a` or `P|b`
B. `P|(a+b)`
C. `P|(a-b)`
D. None of these

Answer

To solve the problem, we need to analyze the statement: If \( P \) is a prime number and \( P \) divides \( ab \) (i.e., \( P | (ab) \)), where \( a \) and \( b \) are integers, then we need to determine which of the given options is correct.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We know that \( P \) is a prime number.
- We are given that \( P | (ab) \), which means that \( ab \) is divisible by \( P \).

2. **Applying the Theorem**:
- There is a well-known theorem in number theory that states: If \( P \) is a prime number and \( P | (ab) \), then \( P | a \) or \( P | b \).
- This means that if \( P \) divides the product \( ab \), it must divide at least one of the factors \( a \) or \( b \).

3. **Evaluating the Options**:
- **Option A**: \( P | a \) or \( P | b \) — This is consistent with the theorem we just stated.
- **Option B**: \( P | (a + b) \) — This is not necessarily true. For example, if \( a = 2 \) and \( b = 3 \) and \( P = 5 \), then \( ab = 6 \) (not divisible by 5), but \( a + b = 5 \) (divisible by 5) does not hold in general.
- **Option C**: \( P | (a - b) \) — Similar reasoning applies here. For \( a = 2 \) and \( b = 3 \), \( a - b = -1 \) (not divisible by 5).
- **Option D**: None of these — This option is incorrect since we have established that option A is correct.

4. **Conclusion**:
- The correct answer is **Option A**: \( P | a \) or \( P | b \).