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Appearance. In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is ...
The greatest common divisor is not unique: if d is a GCD of p and q, then the polynomial f is another GCD if and only if there is an invertible element u of F such that = and =. In other words, the GCD is unique up to the multiplication by an invertible constant.
The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c. [6] The greatest common divisor can be visualized as follows. [7] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly.
In arithmetic and number theory, the least common multiple, 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 divisible by both a and b. [1][2] Since division of integers by zero is undefined, this definition has meaning only if a and b are both ...
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, [1] [2] is an algorithm that computes the greatest common divisor (GCD) of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm ; it replaces division with arithmetic shifts ...
Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a quotient Q and a remainder R such that. and either R = 0 or the degree of R is lower than the degree of B. These conditions uniquely define Q and R ...
As () is a repeated factor, we now need to find two numbers, as so we need an additional relation in order to solve for both. To write the relation of numerators the second fraction needs another factor of ( 1 − 2 x ) {\displaystyle (1-2x)} to convert it to the LCD, giving us 3 x + 5 = A + B ( 1 − 2 x ) {\displaystyle 3x+5=A+B(1-2x)} .
where both A n (z) and B n (z) have integer coefficients, A n (z) has degree φ(n)/2, and B n (z) has degree φ(n)/2 − 2. Furthermore, A n (z) is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, B n (z) is palindromic unless n is composite and ≡ 3 (mod 4), in which case it is antipalindromic.