Find the HCF of 867 and 255, using Euclid's division algorithm.

Answer

To find the HCF of 867 and 255 using Euclid's division algorithm, we will follow these steps:

### Step 1: Apply Euclid's Division Algorithm
According to Euclid's division algorithm, we can express the relationship between two numbers \( a \) and \( b \) (where \( a > b \)) as:
\[ a = bq + r \]
where:
- \( a \) is the larger number (867),
- \( b \) is the smaller number (255),
- \( q \) is the quotient,
- \( r \) is the remainder.

We start by dividing 867 by 255.

### Calculation:
1. Divide 867 by 255:
\[ 867 \div 255 = 3 \quad \text{(quotient)} \]
Now, calculate the remainder:
\[ 867 - (255 \times 3) = 867 - 765 = 102 \]
So, we can express this as:
\[ 867 = 255 \times 3 + 102 \]

### Step 2: Repeat the Process
Now, we will apply the same process to the divisor (255) and the remainder (102).

2. Divide 255 by 102:
\[ 255 \div 102 = 2 \quad \text{(quotient)} \]
Calculate the remainder:
\[ 255 - (102 \times 2) = 255 - 204 = 51 \]
So, we can express this as:
\[ 255 = 102 \times 2 + 51 \]

### Step 3: Continue Until Remainder is Zero
Now, we will apply the process again to the last divisor (102) and the last remainder (51).

3. Divide 102 by 51:
\[ 102 \div 51 = 2 \quad \text{(quotient)} \]
Calculate the remainder:
\[ 102 - (51 \times 2) = 102 - 102 = 0 \]
So, we can express this as:
\[ 102 = 51 \times 2 + 0 \]

### Step 4: Conclusion
Since the remainder is now 0, the last non-zero remainder (51) is the HCF of 867 and 255.

Thus, the HCF of 867 and 255 is:
\[ \text{HCF} = 51 \]

---