The HCF of 144, 180 and 192 is

Options

A. 12
B. 16
C. 18
D. 8

Answer

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

### Step 1: Find the Prime Factorization of Each Number

**For 144:**
1. Divide by 2: \(144 \div 2 = 72\)
2. Divide by 2: \(72 \div 2 = 36\)
3. Divide by 2: \(36 \div 2 = 18\)
4. Divide by 2: \(18 \div 2 = 9\)
5. Divide by 3: \(9 \div 3 = 3\)
6. Divide by 3: \(3 \div 3 = 1\)

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

**For 180:**
1. Divide by 2: \(180 \div 2 = 90\)
2. Divide by 2: \(90 \div 2 = 45\)
3. Divide by 3: \(45 \div 3 = 15\)
4. Divide by 3: \(15 \div 3 = 5\)
5. Divide by 5: \(5 \div 5 = 1\)

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

**For 192:**
1. Divide by 2: \(192 \div 2 = 96\)
2. Divide by 2: \(96 \div 2 = 48\)
3. Divide by 2: \(48 \div 2 = 24\)
4. Divide by 2: \(24 \div 2 = 12\)
5. Divide by 2: \(12 \div 2 = 6\)
6. Divide by 2: \(6 \div 2 = 3\)
7. Divide by 3: \(3 \div 3 = 1\)

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

### Step 2: 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 3: Calculate the HCF

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

### Conclusion

The HCF of 144, 180, and 192 is **12**.

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