Find the HCF os 180,252 and 324 by prime factorization method.

Answer

To find the HCF (Highest Common Factor) of the numbers 180, 252, and 324 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 180
1. Start by dividing 180 by the smallest prime number, which is 2:
\[ 180 \div 2 = 90 \]
2. Divide 90 by 2 again:
\[ 90 \div 2 = 45 \]
3. Now, divide 45 by the next smallest prime number, which is 3:
\[ 45 \div 3 = 15 \]
4. Divide 15 by 3 again:
\[ 15 \div 3 = 5 \]
5. Finally, divide 5 by 5:
\[ 5 \div 5 = 1 \]

Now, we can express 180 as a product of its prime factors:
\[ 180 = 2^2 \times 3^2 \times 5^1 \]

### Step 2: Prime Factorization of 252
1. Start by dividing 252 by 2:
\[ 252 \div 2 = 126 \]
2. Divide 126 by 2 again:
\[ 126 \div 2 = 63 \]
3. Now, divide 63 by 3:
\[ 63 \div 3 = 21 \]
4. Divide 21 by 3 again:
\[ 21 \div 3 = 7 \]
5. Finally, divide 7 by 7:
\[ 7 \div 7 = 1 \]

Now, we can express 252 as a product of its prime factors:
\[ 252 = 2^2 \times 3^2 \times 7^1 \]

### Step 3: Prime Factorization of 324
1. Start by dividing 324 by 2:
\[ 324 \div 2 = 162 \]
2. Divide 162 by 2 again:
\[ 162 \div 2 = 81 \]
3. Now, divide 81 by 3:
\[ 81 \div 3 = 27 \]
4. Divide 27 by 3 again:
\[ 27 \div 3 = 9 \]
5. Divide 9 by 3 again:
\[ 9 \div 3 = 3 \]
6. Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]

Now, we can express 324 as a product of its prime factors:
\[ 324 = 2^2 \times 3^4 \]

### Step 4: Finding the HCF
Now that we have the prime factorization of all three numbers:
- \(180 = 2^2 \times 3^2 \times 5^1\)
- \(252 = 2^2 \times 3^2 \times 7^1\)
- \(324 = 2^2 \times 3^4\)

To find the HCF, we take the lowest power of each common prime factor:
- For the prime factor 2: The lowest power is \(2^2\).
- For the prime factor 3: The lowest power is \(3^2\).
- The prime factors 5 and 7 are not common to all three numbers.

Now we multiply the common prime factors:
\[ HCF = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Final Answer
The HCF of 180, 252, and 324 is **36**.

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