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The Euclidean division algorithm, also known as Euclid’s algorithm, is a method for finding the greatest common divisor (GCD) of two integers. It is based on the principle that the GCD of two numbers remains the same when the larger number is divided by the smaller number. The algorithm involves successive divisions until a remainder of zero is obtained, at which point the divisor of the previous division is the GCD.

Here’s a step-by-step explanation of the Euclidean division algorithm:

1. Start with two positive integers, let’s call them “a” and “b”, where a > b.
2. Divide “a” by “b” and obtain the remainder. Let’s denote the quotient as “q” and the remainder as “r”. The equation can be written as:
   a = bq + r
3. If the remainder “r” is zero, then the algorithm terminates, and the GCD is equal to “b”. In this case, “b” is the largest common divisor of “a” and “b”.
4. If the remainder “r” is not zero, replace “a” with “b” and “b” with “r”. Then, go back to step 2 and repeat the process.
5. Continue the process of dividing the divisor by the remainder until the remainder becomes zero. The last non-zero remainder is the GCD of the original two numbers “a” and “b”.

In mathematical terms, the algorithm can be summarized as follows:
GCD(a, b) = GCD(b, a mod b)

The Euclidean division algorithm is a fundamental method for finding the GCD and has various applications, such as simplifying fractions, checking coprimality (two numbers are coprime if their GCD is 1), and solving linear Diophantine equations.