Extended Euclid's algorithm
Euclid's algorithm
Euclidean
- Computing the greatest common divisor of two numbers
\(\text{gcd}(a,\,b) = \text{gcd}(b,\,a\%b)\)
exEuclid algorithm
- 裴蜀定理：\(\forall a,\,b,\,\,\exists x,\,y, \,\,s.t. \,\,ax+by=\gcd(a,b)\)
- It can be calculated using the extended Euclid algorithm \ (X \) and \ (Y \)
- \ (b = 0 \) , the common factor is \ (A \)
- \ (b \ not = 0 \ ) when, according to the Euclid algorithm, \ (\ text {gcd} ( a, \, b) = \ text {gcd} (b, \, a \% b) \)
- Shu Pei using Theorem: \ (= AX + BX by '+ (A \% B) Y' \)
- 而\(a\%b=a-b\lfloor\frac{a}{b}\rfloor\)
- That is to say: \ (= AX + BX by '+ (ab & \ lfloor \ FRAC {A} {B} \ rfloor) Y' \)
- Sort out: ax + by = ay '+ b (x' - \ lfloor \ frac {a} {b} \ rfloor y ')
- Can be obtained from the above \ (X = Y '\) , \ (Y = X' - \ lfloor \ FRAC {A} {B} \ rfloor Y '\)
- Then you can write a recursive it!
int exgcd(int a, int b, int &x, int &y){
if(b == 0){
x = 1;
y = 0;
return a;
}else{
int d = exgcd(b, a % b, x, y);
int tmp = x;
x = y;
y = tmp - a / b * y;
return d;
}
}