Find the HCF of he numbers in each of the following , using the prime factorization method :
72, 108, 180

Answer

To find the HCF (Highest Common Factor) of the numbers 72, 108, and 180 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of Each Number

**Prime Factorization of 72:**
1. Divide 72 by the smallest prime number, which is 2:
- \( 72 \div 2 = 36 \)
2. Divide 36 by 2:
- \( 36 \div 2 = 18 \)
3. Divide 18 by 2:
- \( 18 \div 2 = 9 \)
4. Now, 9 cannot be divided by 2, so we move to the next prime number, which is 3:
- \( 9 \div 3 = 3 \)
5. Finally, divide 3 by 3:
- \( 3 \div 3 = 1 \)

Thus, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

**Prime Factorization of 108:**
1. Divide 108 by 2:
- \( 108 \div 2 = 54 \)
2. Divide 54 by 2:
- \( 54 \div 2 = 27 \)
3. Now, 27 cannot be divided by 2, so we move to 3:
- \( 27 \div 3 = 9 \)
4. Divide 9 by 3:
- \( 9 \div 3 = 3 \)
5. Finally, divide 3 by 3:
- \( 3 \div 3 = 1 \)

Thus, the prime factorization of 108 is:
\[ 108 = 2^2 \times 3^3 \]

**Prime Factorization of 180:**
1. Divide 180 by 2:
- \( 180 \div 2 = 90 \)
2. Divide 90 by 2:
- \( 90 \div 2 = 45 \)
3. Now, 45 cannot be divided by 2, so we move to 3:
- \( 45 \div 3 = 15 \)
4. Divide 15 by 3:
- \( 15 \div 3 = 5 \)
5. Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

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

### Step 2: Identify Common Factors
Now we will identify the common prime factors from the factorizations:
- \( 72 = 2^3 \times 3^2 \)
- \( 108 = 2^2 \times 3^3 \)
- \( 180 = 2^2 \times 3^2 \times 5^1 \)

The common prime factors are:
- For 2: The minimum power is \( 2^2 \) (common in 72, 108, and 180)
- For 3: The minimum power is \( 3^2 \) (common in 72 and 180, and 108 has \( 3^3 \))

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

Calculating this gives:
\[ 2^2 = 4 \]
\[ 3^2 = 9 \]
\[ \text{HCF} = 4 \times 9 = 36 \]

### Final Answer
The HCF of 72, 108, and 180 is **36**.

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