The H.C.F. of 144 and 198 is

Options

A. 6
B. 9
C. 12
D. 18

Answer

To find the Highest Common Factor (H.C.F.) of 144 and 198, we will follow these steps:

### Step 1: Prime Factorization of 144
We start by finding the prime factors of 144.

1. Divide 144 by 2 (the smallest prime number):
- 144 ÷ 2 = 72
2. Divide 72 by 2:
- 72 ÷ 2 = 36
3. Divide 36 by 2:
- 36 ÷ 2 = 18
4. Divide 18 by 2:
- 18 ÷ 2 = 9
5. Divide 9 by 3:
- 9 ÷ 3 = 3
6. Divide 3 by 3:
- 3 ÷ 3 = 1

Now we can write the prime factorization of 144:
- \( 144 = 2^4 \times 3^2 \)

### Step 2: Prime Factorization of 198
Next, we find the prime factors of 198.

1. Divide 198 by 2:
- 198 ÷ 2 = 99
2. Divide 99 by 3:
- 99 ÷ 3 = 33
3. Divide 33 by 3:
- 33 ÷ 3 = 11
4. Divide 11 by 11:
- 11 ÷ 11 = 1

Now we can write the prime factorization of 198:
- \( 198 = 2^1 \times 3^2 \times 11^1 \)

### Step 3: Identify Common Prime Factors
Now we will identify the common prime factors from both factorizations:

- For 144: \( 2^4 \times 3^2 \)
- For 198: \( 2^1 \times 3^2 \times 11^1 \)

The common prime factors are:
- \( 2^1 \) (the minimum power of 2)
- \( 3^2 \) (the minimum power of 3)

### Step 4: Calculate H.C.F.
Now we multiply the common prime factors:
- H.C.F. = \( 2^1 \times 3^2 = 2 \times 9 = 18 \)

### Conclusion
Thus, the H.C.F. of 144 and 198 is **18**.

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