Extended Euclidean algorithm

Extended Euclidean algorithm uses

When we know $a,b$

Extended Euclidean algorithm can find a $ (x, y) $ solution that satisfies $a*x+b*y=gcd (A, b) $

$GCD (A, b) $ represents $a,b$ greatest common divisor

Preamble Knowledge

$GCD (A, B) =gcd (B,A\%B) $

$GCD (a,0) =0$

$a \%b=a-a/b*b$

Derivation Process

Actually, it's natural to extend Euclid's derivation process.

$a *x+b*y$

$=GCD (A, b) $

$=GCD (b,a\%b) $

$=b*x+ (a\%b) *y$

$=b*x+ (a-a/b*b) *y$

$=b*x+a*y-a/b*b*y$

$=a*y+b*x-a/b*b*y$

$=a*y+ (x-y*a/b) *b$

It's going to go on and on.

When $b=0$

$x =1,y=0$

Code

Attention:

We need to use the $x$ on the previous level when we are asking for $ (x-y*a/b) $

But at this point the previous layer of $x$ has been assigned to $y$

So we need to open an intermediate variable to record the $x$ on the previous layer.

int exgcd (int a,int b,int &x,int &y) {    if (b==0)    {        x=1,y=0;        return A;    }    int R=EXGCD (b,a%b,x,y), TMP;    Tmp=x,x=y,y=tmp-a/b*y;    return r;}

Application 1

The most important application of expanding Euclid is to find the solution of the shape as $a*x+b*y=c$

So how do you ask for it?

First of all, this equation is capable of $c\%gcd (A, B) =0$, which should be more obvious

According to the extended Euclidean algorithm that was previously

We can first find out the solution of $A*X_0+B*Y_0=GCD (A, b) $ $x_0,y_0$

Then divide both sides of the equation by $GCD (A, b) $

The solution of $A*X_0/GCD (A, B) +b*y_0/gcd (A, B) =1$ is obtained.

and multiply $c$ on both sides of the equation.

We get the equation.

$a *x_0/gcd (A, B) *c+b*y_0/gcd (A, b) *c=c$

Isn't it simple?

2

If $GCD (A, B) =1$, and $x0,y0$ is a group of solutions for $a*x+b*y=c$, then any one solution of the equation can be expressed as

$x =x_0+b*t,y=y_0-a*t$

Prove:

$a *x+b*y$

$=a* (x_0+b*t) +b* (y_0-a*t) $

$=a*x_0+a*b*t+b*y_0-a*b*t$

$=a*x_0+b*y_0$

Examples

Rokua P1516 The frog's date

The equations are listed according to the topic requirements, simplifying can be

Exercises

Extended Euclidean algorithm