The Euclidean algorithm is a method used to find the highest common factor (HCF) of two given numbers. It involves repeatedly dividing the larger number by the smaller number, taking the remainder each time, until the remainder becomes zero. The last non-zero remainder obtained is the HCF of the two numbers.

Let's use Euclid's division algorithm to find the HCF of the given numbers.

(i) HCF of 196 and 38220:
Step 1: Divide the larger number by the smaller number and find the remainder.
38220 ÷ 196 = 195 remainder 90

Step 2: Divide the previous divisor (196) by the remainder (90) and find the new remainder.
196 ÷ 90 = 2 remainder 16

Step 3: Repeat step 2 until the remainder becomes zero.
90 ÷ 16 = 5 remainder 10
16 ÷ 10 = 1 remainder 6
10 ÷ 6 = 1 remainder 4
6 ÷ 4 = 1 remainder 2
4 ÷ 2 = 2 remainder 0

Step 4: The last non-zero remainder obtained is 2. Therefore, the HCF of 196 and 38220 is 2.

(ii) HCF of 867 and 255:
Step 1: Divide the larger number by the smaller number and find the remainder.
867 ÷ 255 = 3 remainder 102

Step 2: Divide the previous divisor (255) by the remainder (102) and find the new remainder.
255 ÷ 102 = 2 remainder 51

Step 3: Repeat step 2 until the remainder becomes zero.
102 ÷ 51 = 2 remainder 0

Step 4: The last non-zero remainder obtained is 51. Therefore, the HCF of 867 and 255 is 51.

In both cases, we applied Euclid's division algorithm by dividing the larger number by the smaller number and finding the remainder. Then, we continued to divide the previous divisor by the remainder until the remainder became zero. The last non-zero remainder obtained is the HCF of the given numbers.