Why does the Euclidean algorithm terminate when repeatedly replacing the pair?

The greatest common divisor of 10 and 15 is 5. This is determined by listing the common positive divisors of both numbers, which are 1 and 5, and selecting the largest one.

Conditions: The inputs are the positive integers 10 and 15.; The method used is listing all common positive divisors.

The Euclidean algorithm keeps the GCD by repeatedly applying the step $\gcd(a,b)=\gcd(b,a\bmod b)$ for nonnegative $a$ and positive $b$. This replaces the pair with the divisor and the remainder, preserving the greatest common divisor.

Conditions: $a$ is a nonnegative integer.; $b$ is a positive integer.; The operation $a\bmod b$ computes the remainder of $a$ divided by $b$.

Yes, in this context, "greatest common factor" is being used for what is more commonly called the greatest common divisor in many modern texts. The mathematical procedure shown is the same subtraction-based Euclidean algorithm.

Conditions: Used informally in the explanation of why the algorithm works.

To find the GCF by listing factors, list all positive factors of the first number, list all positive factors of the second number, identify the factors that appear in both lists, and select the largest number from the common factors.

Conditions: The inputs are positive integers.; Listing is practical for the small examples; it is not asserted to be the fastest method.

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 $0 \le r < 10$.