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For integers a and b that are not both zero, the greatest common divisor is the unique positive integer dividing both that is divisible by every other common divisor. Signs do not change its positive value.
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From 45 to 180 seconds, the lesson lists the positive factors of 12 and 42, identifies the common factors 1,2,3,6, and selects 6 as the greatest. Editorial notes make the positive-integer scope explicit.
From 7 to 31 seconds, the source defines the greatest common divisor as the largest integer dividing both inputs and contrasts direct factor testing with the need for an efficient method.
From 34 to 245 seconds, the board correctly computes gcd(10,45)=5 and gcd(1701,3768)=3 using complete division chains. The public review corrects the speaker's repeated phrase greatest common denominator to greatest common divisor.
From 180 to 250 seconds, the source compares dividing by 2 with dividing by the greatest common factor 6, reaching the fully reduced fraction .
From 143 to 163 seconds, the final line is , the last nonzero remainder is 1, and the board records gcd(5295,4321)=1; the speaker concludes that the pair is relatively prime.
From 14 to 44 seconds, the board correctly shows and , ending with the greatest common factor 4.
From 100 to 122 seconds, the lesson reaches , identifies 21 as the last nonzero remainder, and concludes gcd(1785,546)=21.
A reviewed path from functions and finite counting to arrangements, selections, divisibility and the Euclidean algorithm. Every new concept is paired with verified video evidence, while the prerequisite arrows describe this map’s editorial learning order.
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 , 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 .
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 .
To find the greatest common divisor of two large numbers, repeatedly apply the division-with-remainder step. Start by dividing the larger number by the smaller number.
Conditions: The inputs are two positive integers.; The division algorithm is applied at each step.; The process stops when a remainder equals 0.
The speaker verbally says "greatest common denominator," but the mathematical notation on the board is "gcd," which conventionally stands for "greatest common divisor." The context of dividing integers to find a common factor confirms that the intended concept is the greatest common divisor, and the spoken word is a verbal slip.
Conditions: The video discusses finding the common factor of two integers.; The board displays the notation gcd(a;b).; The procedure involves repeated integer division.
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 example ends with 4 because applying the subtraction rule repeatedly yields 4 as the final value. First, , creating the pair 8 and 4.
Conditions: Start with the pair 12 and 8.; Repeatedly subtract the smaller from the larger.
Knowing the greatest common factor (GCF) of 12 and 42, which is 6, allows you to divide both the numerator and the denominator by 6 directly. This immediately yields the simplest form , bypassing intermediate steps like dividing by 2 first.
Conditions: The fraction is .; The GCF of 12 and 42 is known to be 6.
It means that the numerator 2 and the denominator 7 have no common positive divisor greater than 1. Their greatest common factor is 1, so the fraction cannot be reduced any further.
Conditions: The fraction is .; The numerator and denominator are positive integers.
To compute , apply the Euclidean algorithm by repeatedly dividing the previous divisor by the previous remainder. Start with .
Conditions: The inputs are 5295 and 4321.; The Euclidean algorithm is used.; The division algorithm is applied at each step.
The Euclidean algorithm computes the greatest common divisor by repeatedly applying the division algorithm. Starting with two natural numbers and , you divide the larger by the smaller to get a quotient and a remainder.
Conditions: The inputs and are natural numbers.; The division algorithm is applied repeatedly.; The process stops when a remainder equals 0.