Find the HCF and the LCM of 1152 and 1664.

Answer

To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of 1152 and 1664, we can follow these steps:

### Step 1: Find the HCF using the Division Method
1. **Divide the larger number by the smaller number.**
- Divide 1664 by 1152.
- \( 1664 \div 1152 = 1 \) (quotient) with a remainder of \( 1664 - 1152 = 512 \).

2. **Now, take the divisor (1152) and the remainder (512) and repeat the process.**
- Divide 1152 by 512.
- \( 1152 \div 512 = 2 \) (quotient) with a remainder of \( 1152 - (512 \times 2) = 128 \).

3. **Repeat again with the new divisor (512) and the new remainder (128).**
- Divide 512 by 128.
- \( 512 \div 128 = 4 \) (quotient) with a remainder of \( 512 - (128 \times 4) = 0 \).

4. **When the remainder is 0, the last divisor is the HCF.**
- Here, the last divisor is 128.
- Therefore, **HCF(1152, 1664) = 128**.

### Step 2: Find the LCM using the relationship between HCF and LCM
1. **Use the formula:**
\[ \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \]
- The product of 1152 and 1664 is:
\[ 1152 \times 1664 = 1923072 \]

2. **Now, substitute the HCF into the formula to find the LCM:**
\[ 128 \times \text{LCM} = 1923072 \]

3. **Solve for LCM:**
\[ \text{LCM} = \frac{1923072}{128} = 15000 \]

4. **Thus, the LCM of 1152 and 1664 is:**
- **LCM(1152, 1664) = 15000**.

### Final Answer:
- HCF = 128
- LCM = 15000

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