Find the HCF of the numbers in each of the following, using the division method :
754, 1508, 1972

Answer

To find the HCF (Highest Common Factor) of the numbers 754, 1508, and 1972 using the division method, we will follow these steps:

### Step-by-Step Solution:

1. **Divide the largest number by the smallest number:**
- Start by dividing 1972 by 754.
- Calculation:
\[ 1972 \div 754 = 2 \quad \text{(quotient)} \quad \text{with a remainder of } 464 \]
- So, we write:
\[ 1972 = 754 \times 2 + 464 \]

2. **Now, take the divisor (754) and the remainder (464) and repeat the process:**
- Divide 754 by 464.
- Calculation:
\[ 754 \div 464 = 1 \quad \text{(quotient)} \quad \text{with a remainder of } 290 \]
- So, we write:
\[ 754 = 464 \times 1 + 290 \]

3. **Continue with the last divisor (464) and the new remainder (290):**
- Divide 464 by 290.
- Calculation:
\[ 464 \div 290 = 1 \quad \text{(quotient)} \quad \text{with a remainder of } 174 \]
- So, we write:
\[ 464 = 290 \times 1 + 174 \]

4. **Now take 290 and 174:**
- Divide 290 by 174.
- Calculation:
\[ 290 \div 174 = 1 \quad \text{(quotient)} \quad \text{with a remainder of } 116 \]
- So, we write:
\[ 290 = 174 \times 1 + 116 \]

5. **Next, divide 174 by 116:**
- Calculation:
\[ 174 \div 116 = 1 \quad \text{(quotient)} \quad \text{with a remainder of } 58 \]
- So, we write:
\[ 174 = 116 \times 1 + 58 \]

6. **Now divide 116 by 58:**
- Calculation:
\[ 116 \div 58 = 2 \quad \text{(quotient)} \quad \text{with a remainder of } 0 \]
- So, we write:
\[ 116 = 58 \times 2 + 0 \]

7. **Since the remainder is now 0, the divisor at this step (58) is the HCF of 754 and 1972.**
- So, HCF(754, 1972) = 58.

8. **Now, we need to check if 58 is also the HCF with the third number (1508):**
- Divide 1508 by 58.
- Calculation:
\[ 1508 \div 58 = 26 \quad \text{(quotient)} \quad \text{with a remainder of } 0 \]
- So, we write:
\[ 1508 = 58 \times 26 + 0 \]

9. **Since the remainder is 0, we confirm that 58 is also the HCF of 1508.**

### Conclusion:
The HCF of the numbers 754, 1508, and 1972 is **58**.

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