Euclidean Algorithm Back Substitution Calculator
Tool to apply the extended GCD algorithm (Euclidean method) in social club to find the values of the Bezout coefficients and the value of the GCD of 2 numbers.
Extended GCD Algorithm - dCode
Tag(south) : Arithmetics
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Extended GCD Algorithm
- Mathematics
- Arithmetic
- Extended GCD Algorithm
Extended GCD Reckoner
Answers to Questions (FAQ)
What is Extended GCD algorithm? (Definition)
The extended Euclidean algorithm is a modification of the classical GCD algorithm allowing to find a linear combination. From 2 natural inegers a and b, its steps allow to summate their GCD and their Bézout coefficients (encounter the identity of Bezout).
Example: $ a=12 $ and $ b=thirty $, thus $ gcd(12, 30) = 6 $
$$ 12 \times -10 + xxx \times iii = six \\ 12 \times -3 + 30 \times 1 = six \\ 12 \times iv + xxx \times -1 = 6 \\ 12 \times 11 + thirty \times -3 = 6 \\ 12 \times xviii + 30 \times -5 = half dozen \\ 12 \times −2+30 \times 1 = 6 $$
How to lawmaking the Extended GCD algorithm?
Here is an eGCD implementation of the pseudo-code algorithm to find the linear combination gcd(a,b) = a.u+b.v:
function extended_gcd(a, b) {// a, b natural integers a < b
r1 = b, r2 = a, u1 = 0, v1 = ane, u2 = one, v2 = 0
while (r2! = 0) do
q = r1 ÷ r2 (integer sectionalisation)
r3 = r1, u3 = u1, v3 = v1,
r1 = r2, u1 = u2, v1 = v2,
r2 = r3 - q * r2, u2 = u3 - q * u2, v2 = v3 - q * v2
end while
return (r1, u1, v1) (r1 natural integer and u1, v1 rational integers)
The values are such that r1 = pgcd(a, b) = a * u1 + b * v1
How does Extended GCD algorithm work with negative numbers?
Using the absolute values for a and b, the balance of the adding is identical thanks to the belongings: $$ a(\text{sign}(a)\cdot x)+b(\text{sign}(b)\cdot y)=one $$
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Extended GCD Algorithm on dCode.fr [online website], retrieved on 2022-10-25,
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Euclidean Algorithm Back Substitution Calculator,
Source: https://www.dcode.fr/extended-gcd
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