Find the HCF of 180, 252 and 324 by Euclid's Division algorithm.

Answer

To find the HCF (Highest Common Factor) of the numbers 180, 252, and 324 using Euclid's Division Algorithm, we will follow these steps:

### Step 1: Find HCF of 324 and 252
1. Let \( A = 324 \) and \( B = 252 \).
2. Apply Euclid's Division Algorithm: \( A = BQ + R \).
- Divide 324 by 252:
- \( 324 \div 252 = 1 \) (quotient)
- Remainder \( R = 324 - (252 \times 1) = 324 - 252 = 72 \).
3. Rewrite the equation: \( 324 = 252 \times 1 + 72 \).

### Step 2: Find HCF of 252 and 72
1. Now, let \( A = 252 \) and \( B = 72 \).
2. Apply Euclid's Division Algorithm:
- Divide 252 by 72:
- \( 252 \div 72 = 3 \) (quotient)
- Remainder \( R = 252 - (72 \times 3) = 252 - 216 = 36 \).
3. Rewrite the equation: \( 252 = 72 \times 3 + 36 \).

### Step 3: Find HCF of 72 and 36
1. Now, let \( A = 72 \) and \( B = 36 \).
2. Apply Euclid's Division Algorithm:
- Divide 72 by 36:
- \( 72 \div 36 = 2 \) (quotient)
- Remainder \( R = 72 - (36 \times 2) = 72 - 72 = 0 \).
3. Rewrite the equation: \( 72 = 36 \times 2 + 0 \).

### Conclusion
Since the remainder is now 0, the last non-zero remainder is the HCF. Therefore, the HCF of 252 and 324 is 36.

### Step 4: Find HCF of 180 and 36
1. Now, we need to find the HCF of 180 and 36.
2. Let \( A = 180 \) and \( B = 36 \).
3. Apply Euclid's Division Algorithm:
- Divide 180 by 36:
- \( 180 \div 36 = 5 \) (quotient)
- Remainder \( R = 180 - (36 \times 5) = 180 - 180 = 0 \).
4. Rewrite the equation: \( 180 = 36 \times 5 + 0 \).

### Final Result
Since the remainder is 0, the last non-zero remainder is the HCF. Therefore, the HCF of 180, 252, and 324 is **36**.

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