Find the H.C.F. of the given numbers by division method:
(i) 15, 48 `" "` (ii) 198, 429 `" "` (iii) 20, 64, 100 `" "` (iv) 120, 144, 204

Answer

To find the H.C.F. (Highest Common Factor) of the given numbers using the division method, we will solve each part step by step.

### (i) Find H.C.F. of 15 and 48

1. **Divide the larger number by the smaller number**:
- Divide 48 by 15.
- 48 ÷ 15 = 3 (since 15 × 3 = 45).
- Remainder = 48 - 45 = 3.

2. **Now divide the previous divisor by the remainder**:
- Divide 15 by 3.
- 15 ÷ 3 = 5 (since 3 × 5 = 15).
- Remainder = 0.

3. **Since the remainder is 0, the last non-zero remainder is the H.C.F.**:
- H.C.F. = 3.

### (ii) Find H.C.F. of 198 and 429

1. **Divide the larger number by the smaller number**:
- Divide 429 by 198.
- 429 ÷ 198 = 2 (since 198 × 2 = 396).
- Remainder = 429 - 396 = 33.

2. **Now divide the previous divisor by the remainder**:
- Divide 198 by 33.
- 198 ÷ 33 = 6 (since 33 × 6 = 198).
- Remainder = 0.

3. **Since the remainder is 0, the last non-zero remainder is the H.C.F.**:
- H.C.F. = 33.

### (iii) Find H.C.F. of 20, 64, and 100

1. **First, find H.C.F. of 20 and 64**:
- Divide 64 by 20.
- 64 ÷ 20 = 3 (since 20 × 3 = 60).
- Remainder = 64 - 60 = 4.

2. **Now divide the previous divisor by the remainder**:
- Divide 20 by 4.
- 20 ÷ 4 = 5 (since 4 × 5 = 20).
- Remainder = 0.

3. **H.C.F. of 20 and 64 is 4**.

4. **Now find H.C.F. of 4 and 100**:
- Divide 100 by 4.
- 100 ÷ 4 = 25 (since 4 × 25 = 100).
- Remainder = 0.

5. **Since the remainder is 0, the H.C.F. is**:
- H.C.F. = 4.

### (iv) Find H.C.F. of 120, 144, and 204

1. **First, find H.C.F. of 120 and 144**:
- Divide 144 by 120.
- 144 ÷ 120 = 1 (since 120 × 1 = 120).
- Remainder = 144 - 120 = 24.

2. **Now divide the previous divisor by the remainder**:
- Divide 120 by 24.
- 120 ÷ 24 = 5 (since 24 × 5 = 120).
- Remainder = 0.

3. **H.C.F. of 120 and 144 is 24**.

4. **Now find H.C.F. of 24 and 204**:
- Divide 204 by 24.
- 204 ÷ 24 = 8 (since 24 × 8 = 192).
- Remainder = 204 - 192 = 12.

5. **Now divide the previous divisor by the remainder**:
- Divide 24 by 12.
- 24 ÷ 12 = 2 (since 12 × 2 = 24).
- Remainder = 0.

6. **Since the remainder is 0, the H.C.F. is**:
- H.C.F. = 12.

### Summary of H.C.F. Results:
- (i) H.C.F. of 15 and 48 is **3**.
- (ii) H.C.F. of 198 and 429 is **33**.
- (iii) H.C.F. of 20, 64, and 100 is **4**.
- (iv) H.C.F. of 120, 144, and 204 is **12**.