The Euclidean algorithm proof is a mathematical proof that demonstrates the validity of the Euclidean algorithm for finding the greatest common divisor (GCD) of two integers. It shows that the Euclidean algorithm will always terminate with the correct GCD, regardless of the order of the numbers or their size.
The Euclidean algorithm works by repeatedly dividing the larger number by the smaller number and using the remainder as the new smaller number in the next iteration. This process continues until the remainder becomes 0, at which point the previous remainder is the GCD of the original two numbers.
The Euclidean algorithm proof is significant because it provides a rigorous and efficient method for finding the GCD of two numbers. It is also the basis for other important algorithms in number theory and cryptography.
The Euclidean algorithm proof assumes that the numbers being used are positive integers. It also assumes that the GCD of the two numbers is a common divisor of both numbers.
The Euclidean algorithm proof is limited to finding the GCD of two numbers. It cannot be used for finding the GCD of more than two numbers. It also does not work for numbers that are not integers, such as fractions or decimals.