Find the HCF of 144 and 198 by the prime factorization method.

Answer

To find the HCF (Highest Common Factor) of 144 and 198 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 144
- Start by dividing 144 by the smallest prime number, which is 2.
- 144 ÷ 2 = 72
- Divide 72 by 2 again.
- 72 ÷ 2 = 36
- Divide 36 by 2 again.
- 36 ÷ 2 = 18
- Divide 18 by 2 again.
- 18 ÷ 2 = 9
- Now, 9 cannot be divided by 2, so we move to the next prime number, which is 3.
- 9 ÷ 3 = 3
- Finally, divide 3 by 3.
- 3 ÷ 3 = 1

So, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

### Step 2: Prime Factorization of 198
- Start by dividing 198 by 2.
- 198 ÷ 2 = 99
- Next, divide 99 by 3 (the next smallest prime number).
- 99 ÷ 3 = 33
- Divide 33 by 3 again.
- 33 ÷ 3 = 11
- Finally, divide 11 by 11.
- 11 ÷ 11 = 1

So, the prime factorization of 198 is:
\[ 198 = 2^1 \times 3^2 \times 11^1 \]

### Step 3: Identify Common Factors
Now, we compare the prime factorizations of both numbers:
- 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 in both factorizations)
- \( 3^2 \) (the minimum power of 3 in both factorizations)

### Step 4: Calculate the HCF
Now, multiply the common prime factors:
\[ HCF = 2^1 \times 3^2 = 2 \times 9 = 18 \]

Thus, the HCF of 144 and 198 is **18**.

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