"In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers."
The largest number that can divide two or more given numbers without a remainder.
Prime numbers: These are numbers that can only be divided by 1 and themselves. Understanding how to identify prime numbers is important when finding the greatest common factor of two numbers.
Divisibility Rules: These rules are used to determine if a number can be evenly divided by another number. For example, the divisibility rule for 3 is that if the sum of the digits in a number is divisible by 3, then the number itself is divisible by 3.
Factors: These are the numbers that divide evenly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiples: These are the numbers that can be obtained by multiplying a given number by another number. For example, the multiples of 3 are 3, 6, 9, 12, 15, and so on.
Greatest Common Factor (GCF): This is the largest number that divides evenly into two or more numbers. For example, the greatest common factor of 24 and 36 is 12.
Euclidean Algorithm: This is a method used to find the greatest common factor of two numbers by repeatedly subtracting the smaller number from the larger number until one of the numbers becomes zero.
Fraction Reduction: This involves simplifying a fraction by dividing the numerator and denominator by their greatest common factor. For example, the fraction 24/36 can be reduced to 2/3 by dividing both the numerator and denominator by the greatest common factor 12.
LCM and GCF: The Least Common Multiple (LCM) and Greatest Common Factor (GCF) are related concepts, and it is important to understand the relationship between the two in order to find both LCM and GCF of two or more numbers.
Prime Factorization: This is the process of breaking down a number into its prime factors. For example, the prime factorization of 24 is 2 x 2 x 2 x 3.
GCF and Fractions: GCF can be used to simplify fractions by dividing both the numerator and denominator by the GCF. This helps to put fractions in their simplest form.
Euclidean Algorithm: The Euclidean Algorithm is a mathematical method used to find the greatest common factor of two numbers. It involves dividing the larger number by the smaller number and then using the remainder as the new smaller number in a new division. This process is repeated until there is no remainder left, at which point the last divisor used is the greatest common factor.
Prime Factorization: Prime factorization involves finding the prime factors of each of the numbers being considered and then identifying the factors that both numbers have in common. The product of these common factors is the greatest common factor.
Division Method: Division method involves dividing the larger number by the smaller number and then dividing the result by any remainder, repeating the process until the remainder is zero. The last divisor used is the greatest common factor.
Common Multiples Method: Common multiples method involves identifying all of the multiples of both numbers being considered and then identifying the largest multiple that both numbers have in common. This number is the greatest common factor.
Subtraction Method: Subtraction method involves subtracting the smaller number from the larger number and then repeating the process with the new difference and the smaller number until the difference is zero. The last number subtracted is the greatest common factor.
Extended Euclidean Algorithm: The extended Euclidean Algorithm involves finding the greatest common factor of two numbers along with two additional numbers, x and y, such that ax + by = gcd(a, b). This method is useful in cryptography and other applications.
Binary Euclidean Algorithm: The binary Euclidean Algorithm involves finding the greatest common factor of two numbers by repeatedly dividing by 2 and factoring out powers of 2 until the two numbers are odd. The algorithm is then applied recursively until the two numbers are equal, at which point the gcd is equal to one of the numbers times the power of 2 that was factored out.
Continued Fraction Algorithm: Continued fraction algorithm involves finding the greatest common factor of two numbers by expressing the larger number as a continued fraction in terms of the smaller number and then determining the last integer in the fraction. This integer is the greatest common factor.
Bezout's Identity: Bezout's identity states that for any two integers a and b, there exist integers x and y such that ax + by = gcd(a, b). This identity can be used to find the greatest common factor of two numbers along with its associated values of x and y.
Chinese Remainder Theorem: The Chinese remainder theorem involves finding the greatest common factor of two numbers that are relatively prime to each other by solving a set of modular equations. This method is useful in number theory and cryptography.
"For two integers x, y, the greatest common divisor of x and y is denoted gcd(x, y)."
"The GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4."
"The adjective 'greatest' may be replaced by 'highest', and the word 'divisor' may be replaced by 'factor', so that other names include highest common factor (hcf), etc."
"This notion can be extended to polynomials (see Polynomial greatest common divisor)."
"Historically, other names for the same concept have included greatest common measure."
"The greatest common divisor can also be extended to other commutative rings (see ยง In commutative rings below)."
"The paragraph states that the integers should 'not all be zero', so the definition doesn't apply if the dividend and divisor are both zero."
"Yes, the greatest common divisor is the 'largest positive integer that divides each of the integers.'"
"The definition states the 'largest positive integer,' implying that there can only be one greatest common divisor."
"If the greatest common divisor of two integers is 1, it means the two integers are coprime or relatively prime."
"No, the greatest common divisor is a divisor of the integers, so it cannot be larger than them."
"There are no specific restrictions mentioned in the paragraph regarding the values of x and y."
"The paragraph does not mention the possibility of the greatest common divisor being negative, so it is assumed to be positive."
"The definition states that the integers should 'not all be zero,' so the greatest common divisor is undefined in that case."
"The greatest common divisor is the largest positive integer that divides each of the given integers."
"The greatest common divisor is used in various algorithms and mathematical operations involving integers."
"The paragraph specifically refers to the greatest common divisor of integers, so it doesn't apply to non-integer values."
"Yes, the most commonly used notation is gcd(x, y)."
"The paragraph explains that the concept of greatest common divisor can be extended to polynomials and other commutative rings, so the divisor may differ in those contexts."