When does the Euclidean algorithm using division terminate?

The Euclidean algorithm moves the old divisor to the left-hand side (new dividend) and the old remainder to the smaller-number position (new divisor) to recursively reduce the problem. This shift ensures that each subsequent division step operates on smaller numbers while preserving the greatest common divisor of the original pair, continuing until a remainder of zero is reached.

Conditions: The algorithm is applied to two positive integers.; The previous remainder is not zero.; The process continues until a remainder of 0 is obtained.

For a nonnegative integer $n$, the greatest common divisor of $n$ and $0$ is $n$ itself. This serves as the base case for the Euclidean algorithm, terminating the recursion when the second argument becomes zero.

Conditions: $n$ is a nonnegative integer.; The pair is $(n, 0)$.

To form the next division line, you take the divisor from the previous line and make it the dividend of the new line. Then, you take the remainder from the previous line and make it the divisor of the new line.

Conditions: You have just completed a division step in the Euclidean algorithm.; The previous remainder is not 0.

To start the Euclidean algorithm for $\gcd(10,45)$, you write the larger number as the smaller number multiplied by an unknown quotient plus an unknown remainder. Specifically, you set up the division equation $45 = 10 \cdot q + r$.

Conditions: The inputs are positive integers.; The larger number is placed on the left-hand side of the equation.; The quotient is an integer and the remainder satisfies $0 \le r < 10$.

Repeating the subtraction operation reduces the magnitude of the numbers involved while preserving their common divisors. Since the numbers are positive integers, they cannot decrease indefinitely.

Conditions: The initial numbers are positive integers.; The operation is subtracting the smaller from the larger.; The goal is to preserve the gcd while reducing the numbers.