Find the H.C.F. of the given numbers by division method:

Options

A. `10`
B. `12`
C. `14`
D. none of these

Answer

To find the H.C.F. (Highest Common Factor) of the numbers 84, 120, and 156 using the division method, we will follow these steps:

### Step 1: Find H.C.F. of the first two numbers (84 and 120)

1. **Divide the larger number by the smaller number**:
- Divide 120 by 84.
- \( 120 \div 84 = 1 \) (the quotient)
- Remainder = \( 120 - 84 \times 1 = 120 - 84 = 36 \)

2. **Repeat the process**:
- Now, divide the previous divisor (84) by the remainder (36).
- \( 84 \div 36 = 2 \)
- Remainder = \( 84 - 36 \times 2 = 84 - 72 = 12 \)

3. **Continue dividing**:
- Now, divide the previous divisor (36) by the remainder (12).
- \( 36 \div 12 = 3 \)
- Remainder = \( 36 - 12 \times 3 = 36 - 36 = 0 \)

4. **Conclusion for the first two numbers**:
- Since the remainder is now 0, the last non-zero remainder is 12.
- Therefore, H.C.F. of 84 and 120 is **12**.

### Step 2: Find H.C.F. of the result (12) and the third number (156)

1. **Divide 156 by 12**:
- \( 156 \div 12 = 13 \)
- Remainder = \( 156 - 12 \times 13 = 156 - 156 = 0 \)

2. **Conclusion for all three numbers**:
- Since the remainder is 0, the last non-zero remainder is 12.
- Therefore, H.C.F. of 84, 120, and 156 is **12**.

### Final Answer:
The H.C.F. of 84, 120, and 156 is **12**.

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