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function gcd(a, b) if b = 0 return a else return gcd(b, a mod b) (As above, if negative inputs are allowed, or if the mod function may return negative values, the instruction return a must be replaced by return max(a, −a).) For illustration, the gcd(1071, 462) is calculated from the equivalent gcd(462, 1071 mod 462) = gcd(462, 147).
If m is any integer, then gcd(a + m⋅b, b) = gcd(a, b). Equivalently, gcd(a mod b,b) = gcd(a,b). If m is a positive common divisor of a and b, then gcd(a/m, b/m) = gcd(a, b)/m. 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).
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under ...
The rest of the proof is easy and transparent if is a unique factorization domain; thus we give the proof in that case here (and see [note 4] for the proof for the GCD case). If gcd ( cont ( f g ) ) = ( 1 ) {\displaystyle \gcd(\operatorname {cont} (fg))=(1)} , then there is nothing to prove.
Here the greatest common divisor of 0 and 0 is taken to be 0.The integers x and y are called Bézout coefficients for (a, b); they are not unique.A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b/d | and | y | ≤ | a/d |; equality occurs only if one of a and b is a multiple ...
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that
As (a, b) and (b, rem(a,b)) have the same divisors, the set of the common divisors is not changed by Euclid's algorithm and thus all pairs (r i, r i+1) have the same set of common divisors. The common divisors of a and b are thus the common divisors of r k−1 and 0. Thus r k−1 is a GCD of a and b. This not only proves that Euclid's algorithm ...
Lamé's Theorem is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm.Using Fibonacci numbers, he proved in 1844 [1] [2] that when looking for the greatest common divisor (GCD) of two integers a and b, the algorithm finishes in at most 5k steps, where k is the number of digits (decimal) of b.