(3) Greatest common divisor

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1 (3) Greatest common divisor In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4. This notion can be extended to polynomials, see greatest common divisor of two polynomials. Overview Example The number 54 can be expressed as a product of two other integers in several different ways: Thus the divisors of 54 are: Similarly the divisors of 24 are: The numbers that these two lists share in common are the common divisors of 54 and 24: The greatest of these is 6. That is the greatest common divisor of 54 and 24. One writes: Reducing fractions The greatest common divisor of a and b is written as gcd(a, b), or sometimes simply as (a, b). For example, gcd(12, 18) = 6, gcd( 4, 14) = 2. 1

2 Two numbers are called relatively prime, or coprime if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime. Calculation Using Euclid's algorithm A much more efficient method is the Euclidean algorithm, which uses the division algorithm in combination with the observation that the gcd of two numbers also divides their difference. To compute gcd(48,18), divide 48 by 18 to get a quotient of 2 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd. Note that we ignored the quotient in each step except to notice when it reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as: gcd(a,0) = a Which also could be written as gcd(a,0) = a gcd(a,b) = gcd(b,a mod b). Other methods If a and b are not both zero, the greatest common divisor of a and b can be computed by using least common multiple (lcm) of a and b: Properties Every common divisor of a and b is a divisor of gcd(a, b). gcd(a, b), where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form d = a p + b q where p and q are integers. This expression is called Bézout's identity. Numbers p and q like this can be computed with the extended Euclidean algorithm. gcd(a, 0) = a, for a 0, since any number is a divisor of 0, and the greatest divisor of a is a. This is usually used as the base case in the Euclidean algorithm. 2

3 If a divides the product b c, and gcd(a, b) = d, then a/d divides c. If m is a non-negative integer, then gcd(m a, m b) = m gcd(a, b). If m is any integer, then gcd(a + m b, b) = gcd(a, b). If m is a nonzero common divisor of a and b, then gcd(a/m, b/m) = gcd(a, b)/m. The gcd is a multiplicative function in the following sense: if a 1 and a 2 are relatively prime, then gcd(a 1 a 2, b) = gcd(a 1, b) gcd(a 2, b). The gcd is a commutative function: gcd(a, b) = gcd(b, a). The gcd is an associative function: gcd(a, gcd(b, c)) = gcd(gcd(a, b), c). The gcd of three numbers can be computed as gcd(a, b, c) = gcd(gcd(a, b), c), or in some different way by applying commutativity and associativity. This can be extended to any number of numbers. gcd(a, b) is closely related to the least common multiple lcm(a, b): we have gcd(a, b) lcm(a, b) = a b. This formula is often used to compute least common multiples: one first computes the gcd with Euclid's algorithm and then divides the product of the given numbers by their gcd. The following versions of distributivity hold true: gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c)) lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c)). It is useful to define gcd(0, 0) = 0 and lcm(0, 0) = 0 because then the natural numbers become a complete distributive lattice with gcd as meet and lcm as join operation. This extension of the definition is also compatible with the generalization for commutative rings given below. In a Cartesian coordinate system, gcd(a, b) can be interpreted as the number of points with integral coordinates on the straight line joining the points (0, 0) and (a, b), excluding (0, 0). ********************* ************ Least common multiple In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is a multiple of both a and b. [1] It is familiar from grade-school arithmetic as the "lowest common denominator" that must be determined before two fractions can be added. If either a or b is 0, LCM(a, b) is defined to be zero. 3

4 The LCM of more than two integers is also well-defined: it is the smallest number that is an integer multiple of each of them. Example What is the LCM of 4 and 6? Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76 etc. and the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72,... Common multiples of 4 and 6 are simply the numbers that are in both lists: 12, 24, 36, 48, 60, 72,... So the least common multiple of 4 and 6 is the smallest one of those: 12 Applications When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, where the denominator 42 was used because it is the least common multiple of 21 and 6. Computing the least common multiple Reduction by the greatest common divisor The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (GCD): 4

5 This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = a. There are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above, Because gcd(a, b) is a divisor of both a and b, it's more efficient to compute the LCM by dividing before multiplying: This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (overflow in the a b computation). Because gcd(a, b) is a divisor of both a and b, and thus the division will be guaranteed to yield an integer, so the intermediate result can be stored in an integer. Done this way, the previous example becomes: Finding least common multiples by prime factorization The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number. For example: Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5. This knowledge can be used to find the lcm of a set of numbers. Example: Find the value of lcm(8,9,21). First, factor out each number and express it as a product of prime number powers. 5

6 The lcm will be the product of multiplying the highest power in each prime factor category together. Out of the 4 prime factor categories 2, 3, 5, and 7, the highest powers from each are 2 3, 3 2, 5 0, and 7 1. Thus, This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful in illustrating concepts. A simple algorithm This method works as easily for finding the LCM of several integers. Let there be a finite sequence of positive integers X = (x 1, x 2,..., x n ), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X (m) = (x 1 (m), x 2 (m),..., x n (m) ), X (1) = X. The purpose of the examination is to pick up the least (perhaps, one of many) element of the sequence X (m). Assuming x k0 (m) is the selected element, the sequence X (m+1) is defined as x k (m+1) = x k (m), k k 0 x k0 (m+1) = x k0 (m) + x k0. In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X (m) to X (m+1) unchanged. The algorithm stops when all elements in sequence X (m) are equal. Their common value L is exactly LCM(X). (For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.) ************************** ********** 6