The extended Euclidean algorithm is an extension of the Euclidean algorithm, a method for finding the greatest common divisor (GCD) of two integers. It was developed by Euclid in ancient Greece and has been a fundamental tool in number theory for centuries.

The Euclidean algorithm works by repeatedly dividing the larger of the two numbers by the smaller, until the remainder is zero. The GCD is then the last non-zero remainder. For example, to find the GCD of 24 and 18, we can use the Euclidean algorithm as follows:

24 ÷ 18 = 1 remainder 6 18 ÷ 6 = 3 remainder 0

Therefore, the GCD of 24 and 18 is 6.

The extended Euclidean algorithm goes one step further and not only finds the GCD, but also computes the integers x and y such that the GCD can be expressed as a linear combination of the two original numbers: GCD = x * a + y * b. This is known as the Bezout's identity.

To find the Bezout's coefficients x and y using the extended Euclidean algorithm, we start with a and b as the two input numbers and compute the remainder r of a divided by b. We then assign x and y the values of the previous x and y values, respectively. We then repeat the process with b and r until r is zero. The final x and y values are the Bezout's coefficients.

For example, to find the GCD of 24 and 18 and the corresponding Bezout's coefficients using the extended Euclidean algorithm, we can proceed as follows:

24 = 1 * 18 + 6 18 = 3 * 6 + 0

We can then use the equations above to find the values of x and y:

x = -3 y = 1

Therefore, the GCD of 24 and 18 is 6 and it can be expressed as a linear combination of 24 and 18 as follows: 6 = (-3) * 24 + 1 * 18.

The extended Euclidean algorithm has numerous applications in cryptography, computer science, and engineering. It is used to find modular inverses, solve linear Diophantine equations, and compute multiplicative inverses in finite fields. It is also used to factorize integers and to test for the primality of numbers.

In conclusion, the extended Euclidean algorithm is a powerful tool that allows us to find the greatest common divisor of two integers and express it as a linear combination of the two numbers. It has a wide range of applications in various fields and continues to be an important tool in number theory and beyond.