Find the HCF of 30,72 and 432.

Answer

To find the HCF (Highest Common Factor) of the numbers 30, 72, and 432, we will use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of Each Number

1. **Prime Factorization of 30:**
- 30 can be divided by 2:
\[ 30 \div 2 = 15 \]
- Next, 15 can be divided by 3:
\[ 15 \div 3 = 5 \]
- 5 is a prime number.
- Therefore, the prime factorization of 30 is:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

2. **Prime Factorization of 72:**
- 72 can be divided by 2:
\[ 72 \div 2 = 36 \]
- 36 can be divided by 2:
\[ 36 \div 2 = 18 \]
- 18 can be divided by 2:
\[ 18 \div 2 = 9 \]
- 9 can be divided by 3:
\[ 9 \div 3 = 3 \]
- 3 is a prime number.
- Therefore, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

3. **Prime Factorization of 432:**
- 432 can be divided by 2:
\[ 432 \div 2 = 216 \]
- 216 can be divided by 2:
\[ 216 \div 2 = 108 \]
- 108 can be divided by 2:
\[ 108 \div 2 = 54 \]
- 54 can be divided by 2:
\[ 54 \div 2 = 27 \]
- 27 can be divided by 3:
\[ 27 \div 3 = 9 \]
- 9 can be divided by 3:
\[ 9 \div 3 = 3 \]
- 3 is a prime number.
- Therefore, the prime factorization of 432 is:
\[ 432 = 2^4 \times 3^3 \]

### Step 2: Identify the Common Factors

Now, we will identify the common prime factors from the factorizations:

- **For 30:** \(2^1, 3^1\)
- **For 72:** \(2^3, 3^2\)
- **For 432:** \(2^4, 3^3\)

### Step 3: Find the Lowest Power of Common Prime Factors

- For the prime factor 2:
- The lowest power is \(2^1\) (from 30).

- For the prime factor 3:
- The lowest power is \(3^1\) (from 30).

### Step 4: Calculate the HCF

Now, we multiply the lowest powers of the common prime factors:
\[ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Final Answer

The HCF of 30, 72, and 432 is **6**.

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