Using Euclid's division algorithm, find H.C.F. of 274170 and 17017.

Answer

To find the H.C.F. (Highest Common Factor) of 274170 and 17017 using Euclid's division algorithm, we will follow these steps:

### Step 1: Apply Euclid's Division Algorithm
We start by dividing the larger number (274170) by the smaller number (17017).

\[ 274170 \div 17017 \]

Calculating the quotient:

\[ \text{Quotient} = 16 \quad \text{(since } 17017 \times 16 = 272272\text{)} \]

Now, we calculate the remainder:

\[ \text{Remainder} = 274170 - 272272 = 1898 \]

So, we can express this as:

\[ 274170 = 17017 \times 16 + 1898 \]

### Step 2: Repeat the Process
Now, we take the divisor (17017) and the remainder (1898) and apply the division algorithm again.

\[ 17017 \div 1898 \]

Calculating the quotient:

\[ \text{Quotient} = 8 \quad \text{(since } 1898 \times 8 = 15184\text{)} \]

Calculating the remainder:

\[ \text{Remainder} = 17017 - 15184 = 1833 \]

So, we express this as:

\[ 17017 = 1898 \times 8 + 1833 \]

### Step 3: Continue the Process
Next, we take 1898 and 1833:

\[ 1898 \div 1833 \]

Calculating the quotient:

\[ \text{Quotient} = 1 \quad \text{(since } 1833 \times 1 = 1833\text{)} \]

Calculating the remainder:

\[ \text{Remainder} = 1898 - 1833 = 65 \]

So, we express this as:

\[ 1898 = 1833 \times 1 + 65 \]

### Step 4: Continue Again
Now, we take 1833 and 65:

\[ 1833 \div 65 \]

Calculating the quotient:

\[ \text{Quotient} = 28 \quad \text{(since } 65 \times 28 = 1820\text{)} \]

Calculating the remainder:

\[ \text{Remainder} = 1833 - 1820 = 13 \]

So, we express this as:

\[ 1833 = 65 \times 28 + 13 \]

### Step 5: Final Step
Now, we take 65 and 13:

\[ 65 \div 13 \]

Calculating the quotient:

\[ \text{Quotient} = 5 \quad \text{(since } 13 \times 5 = 65\text{)} \]

Calculating the remainder:

\[ \text{Remainder} = 65 - 65 = 0 \]

So, we express this as:

\[ 65 = 13 \times 5 + 0 \]

### Conclusion
Since we have reached a remainder of 0, the last non-zero remainder is the H.C.F.

Thus, the H.C.F. of 274170 and 17017 is **13**.

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