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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.
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
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 ...
compute g = gcd(a M − 1, n) (note: exponentiation can be done modulo n) if 1 < g < n then return g; if g = 1 then select a larger B and go to step 2 or return failure; if g = n then select a smaller B and go to step 2 or return failure; If g = 1 in step 6, this indicates there are no prime factors p for which p-1 is B-powersmooth.
Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, [1] and modular arithmetic, for which only remainders are considered. [2]
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
The greatest common divisor of p and q is usually denoted "gcd(p, q)". 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 f = u d {\displaystyle f=ud} and d = u − 1 f . {\displaystyle d=u^{-1}f.}
mXparser is an open-source mathematical expressions parser/evaluator providing abilities to calculate various expressions at a run time. [1] Expressions definitions are given as plain text, then verified in terms of grammar / syntax, finally calculated.