Find HCF of 144.180 and 192 by the prime factorization method.

Answer

To find the HCF (Highest Common Factor) of 144, 180, and 192 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 144
1. Divide 144 by 2:
\( 144 \div 2 = 72 \)
2. Divide 72 by 2:
\( 72 \div 2 = 36 \)
3. Divide 36 by 2:
\( 36 \div 2 = 18 \)
4. Divide 18 by 2:
\( 18 \div 2 = 9 \)
5. Divide 9 by 3:
\( 9 \div 3 = 3 \)
6. Divide 3 by 3:
\( 3 \div 3 = 1 \)

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

### Step 2: Prime Factorization of 180
1. Divide 180 by 2:
\( 180 \div 2 = 90 \)
2. Divide 90 by 2:
\( 90 \div 2 = 45 \)
3. Divide 45 by 3:
\( 45 \div 3 = 15 \)
4. Divide 15 by 3:
\( 15 \div 3 = 5 \)
5. Divide 5 by 5:
\( 5 \div 5 = 1 \)

Thus, the prime factorization of 180 is:
\[ 180 = 2^2 \times 3^3 \times 5^1 \]

### Step 3: Prime Factorization of 192
1. Divide 192 by 2:
\( 192 \div 2 = 96 \)
2. Divide 96 by 2:
\( 96 \div 2 = 48 \)
3. Divide 48 by 2:
\( 48 \div 2 = 24 \)
4. Divide 24 by 2:
\( 24 \div 2 = 12 \)
5. Divide 12 by 2:
\( 12 \div 2 = 6 \)
6. Divide 6 by 2:
\( 6 \div 2 = 3 \)
7. Divide 3 by 3:
\( 3 \div 3 = 1 \)

Thus, the prime factorization of 192 is:
\[ 192 = 2^6 \times 3^1 \]

### Step 4: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:
- For \( 2 \): The minimum power is \( 2^2 \) (from 180).
- For \( 3 \): The minimum power is \( 3^1 \) (from 192).

### Step 5: Calculate HCF
Now, we multiply the common prime factors:
\[ \text{HCF} = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Conclusion
Thus, the HCF of 144, 180, and 192 is **12**.

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