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The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.

The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.

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.

In the Euclidean algorithm, when the division process yields a remainder of zero, the greatest common divisor of the original two integers is the last non-zero remainder obtained. The algorithm stops at this point because the method is over, and the last non-zero remainder is guaranteed to divide both original numbers evenly.

Conditions: The Euclidean algorithm is applied to two integers.; The process of repeated long division is followed until a remainder of zero is reached.

In the Euclidean algorithm, when the division process yields a remainder of zero, the greatest common divisor of the original two integers is the last non-zero remainder obtained. The algorithm stops at this point because the method is over, and the last non-zero remainder is guaranteed to divide both original numbers evenly.

Conditions: The Euclidean algorithm is applied to two integers.; The process of repeated long division is followed until a remainder of zero is reached.

The greatest common factor is useful for simplifying fractions. Dividing both the numerator and the denominator by their GCF reduces the fraction to its simplest form in one step, where the numerator and denominator have no common positive divisor greater than 1.

Conditions: The fraction has a positive-integer numerator and denominator.; The denominator is nonzero.

The greatest common factor is useful for simplifying fractions. Dividing both the numerator and the denominator by their GCF reduces the fraction to its simplest form in one step, where the numerator and denominator have no common positive divisor greater than 1.

Conditions: The fraction has a positive-integer numerator and denominator.; The denominator is nonzero.

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.

Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.

Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.

Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.

Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.