Use Euclid's division algorithm to find the H.C.F. of 196 and 38220

Answer

To find the H.C.F. of 196 and 38220 using Euclid's division algorithm, we will follow these steps:

### Step 1: Identify the numbers
We have two numbers:
- \( a = 38220 \) (the larger number)
- \( b = 196 \) (the smaller number)

### Step 2: Apply Euclid's Division Algorithm
According to Euclid's division algorithm, we can express \( a \) in terms of \( b \) as follows:
\[ a = bq + r \]
where:
- \( q \) is the quotient,
- \( r \) is the remainder.

We need to divide \( 38220 \) by \( 196 \).

### Step 3: Perform the division
1. Calculate \( 38220 \div 196 \).
- The quotient \( q \) can be calculated as:
\[ q = \lfloor 38220 \div 196 \rfloor = 195 \]
- Now, we calculate the product \( bq \):
\[ bq = 196 \times 195 = 38220 \]
- Now, we can find the remainder \( r \):
\[ r = a - bq = 38220 - 38220 = 0 \]

### Step 4: Check the remainder
Since the remainder \( r = 0 \), we stop here. According to Euclid's algorithm, when the remainder is 0, the divisor \( b \) at this step is the H.C.F.

### Conclusion
Thus, the H.C.F. of 196 and 38220 is:
\[ \text{H.C.F.} = 196 \]