What is the greatest common divisor of 10 and 15 according to the listing method?

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 $0 \le r < 10$.

The Euclidean algorithm terminates because the remainder in each step is strictly smaller than the previous divisor. Since the second argument is a nonnegative integer that decreases with each step, it must eventually reach zero.

Conditions: The algorithm operates on nonnegative integers.; Each step replaces the pair $(a,b)$ with $(b, a\bmod b)$.; The remainder $a\bmod b$ is strictly less than $b$ when $b > 0$.