Consider the following in respect of natural numbers a, b and c -
(1) LCM (ab, ac) = a LCM (b, c)`/`(ab, ac)
(2) HCF (ab, ac) = a HCF (b, c)`/`(ab ac)
(3) HCF (a, b)`gt`LCM (a, b)`/`(a, b)
(4) HCF (a, b) divides LCM (a, b)
Which of the above are correct ?

Options

A. 1 and 2
B. 3 and 4
C. 1, 2 and 4
D. 1, 2, 3 and 4

Answer

To solve the question, we need to analyze the four statements regarding the natural numbers \( a, b, \) and \( c \) one by one.

### Statement 1:
**LCM (ab, ac) = a LCM (b, c)**

1. **Understanding LCM**: The LCM of two numbers \( x \) and \( y \) can be expressed as:
\[ \text{LCM}(x, y) = \frac{x \cdot y}{\text{HCF}(x, y)} \]
2. **Applying to our case**:
- Let \( x = ab \) and \( y = ac \).
- Then, we have:
\[ \text{LCM}(ab, ac) = \frac{(ab)(ac)}{\text{HCF}(ab, ac)} \]
3. **Finding HCF(ab, ac)**:
- The HCF of \( ab \) and \( ac \) is \( a \cdot \text{HCF}(b, c) \).
4. **Substituting back**:
\[ \text{LCM}(ab, ac) = \frac{(ab)(ac)}{a \cdot \text{HCF}(b, c)} = a \cdot \frac{bc}{\text{HCF}(b, c)} = a \cdot \text{LCM}(b, c) \]
5. **Conclusion**: Statement 1 is **correct**.

### Statement 2:
**HCF (ab, ac) = a HCF (b, c)**

1. **Using the HCF definition**:
- We already found that:
\[ \text{HCF}(ab, ac) = a \cdot \text{HCF}(b, c) \]
2. **Conclusion**: Statement 2 is **correct**.

### Statement 3:
**HCF (a, b) > LCM (a, b)**

1. **Understanding HCF and LCM**:
- For any two natural numbers \( a \) and \( b \):
\[ \text{HCF}(a, b) \cdot \text{LCM}(a, b) = a \cdot b \]
2. **Analyzing the inequality**:
- If \( a \) and \( b \) are both positive integers, then \( \text{HCF}(a, b) \) cannot be greater than \( \text{LCM}(a, b) \) since \( a \cdot b \) is always greater than or equal to both \( \text{HCF}(a, b) \) and \( \text{LCM}(a, b) \).
3. **Conclusion**: Statement 3 is **incorrect**.

### Statement 4:
**HCF (a, b) divides LCM (a, b)**

1. **Using the relationship**:
- As stated earlier:
\[ \text{HCF}(a, b) \cdot \text{LCM}(a, b) = a \cdot b \]
2. **Divisibility**:
- Since \( a \cdot b \) is a multiple of \( \text{HCF}(a, b) \), it follows that \( \text{HCF}(a, b) \) divides \( \text{LCM}(a, b) \).
3. **Conclusion**: Statement 4 is **correct**.

### Final Conclusion:
The correct statements are **1, 2, and 4**. Therefore, the answer is that statements 1, 2, and 4 are correct.

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