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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.

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.

The example ends with 4 because applying the subtraction rule repeatedly yields 4 as the final value. First, $12 - 8 = 4$, creating the pair 8 and 4.

Conditions: Start with the pair 12 and 8.; Repeatedly subtract the smaller from the larger.

The example ends with 4 because applying the subtraction rule repeatedly yields 4 as the final value. First, $12 - 8 = 4$, creating the pair 8 and 4.

Conditions: Start with the pair 12 and 8.; Repeatedly subtract the smaller from the larger.

A common divisor also divides the difference because division is interpreted as repeated subtraction. If a number divides evenly into the larger number and the smaller number, subtracting the smaller number repeatedly from the larger one will eventually leave a difference that the same divisor also divides evenly.

Conditions: There are two numbers in the example.; A chosen divisor divides evenly in the sense described by the speaker.; The subtraction is performed from the larger number using the smaller number.

A common divisor also divides the difference because division is interpreted as repeated subtraction. If a number divides evenly into the larger number and the smaller number, subtracting the smaller number repeatedly from the larger one will eventually leave a difference that the same divisor also divides evenly.

Conditions: There are two numbers in the example.; A chosen divisor divides evenly in the sense described by the speaker.; The subtraction is performed from the larger number using the smaller number.