Euclid's Division Algorithm

Euclid's Division Algorithm is a method of finding the highest common factor (HCF) of two given numbers. This algorithm is based on the fact that the HCF of two numbers divides their difference as well as any integer multiple of their difference.

Steps to find HCF using Euclid's Division Algorithm

1. Divide the larger number by the smaller number.
2. Write the remainder as the new dividend and the smaller number as the new divisor.
3. Repeat the above step until the remainder becomes zero.
4. The divisor of the last step is the HCF of the given numbers.

Example: Find HCF of 135 and 225 using Euclid's Division Algorithm

Step 1: Divide the larger number by the smaller number
225 ÷ 135 = 1 with a remainder of 90

Step 2: Write the remainder as the new dividend and the smaller number as the new divisor
135 ÷ 90 = 1 with a remainder of 45

Step 3: Repeat the above step until the remainder becomes zero
90 ÷ 45 = 2 with a remainder of 0

Step 4: The divisor of the last step is the HCF of the given numbers
The HCF of 135 and 225 is 45.

Explanation:
Euclid's Division Algorithm is based on the fact that the HCF of two numbers divides their difference as well as any integer multiple of their difference. In this example, we use this fact to find the HCF of 135 and 225. We divide the larger number (225) by the smaller number (135) to get a quotient of 1 and a remainder of 90. We then use the remainder (90) as the new dividend and the smaller number (135) as the new divisor. We continue this process until the remainder becomes zero. The divisor of the last step (45) is the HCF of the given numbers (135 and 225).