# Euclidean algorithm GCD and its development ultimate explanation

Source: Internet
Author: User

`Extended Euclidean algorithm-solving indefinite equation, linear congruence equation.`
`If the two frogs meet after the S-step, the following equations will be fulfilled:`
`(x+m*s)-(y+n*s) =k*l (k=0,1,2 ...)`
`Slightly changed to form:`
`(n-m) *s+k*l=x-y`
`Make n-m=a,k=b,x-y=c, namely`
`A*s+b*l=c`
`As long as there is an integer solution on the equation, two frogs can meet, or not.`
`The first way to think of a method is to use two times for the loop to enumerate the value of the s,l, see if there is a s,l integer solution, if present, enter the smallest s,`
`But obviously this method is undesirable, and no one knows how big the smallest s is, and if the smallest s is large then the timeout is obvious.`
`In fact, the problem with the Euclidean expansion of the principle can be quickly resolved, first to see what is the Euclidean expansion principle:`
`Euclidean algorithm, also known as the greatest common divisor method, is used to calculate two integers, a, b, and so on. Its computational principle relies on the following theorem:`
`Theorem: gcd (b) = gcd (b,a mod b)`
`Proof: A can be expressed as A = kb + R, then r = a mod b`
`Assuming D is a number of conventions for a, B, there are`
`D|a, d|b, and r = a-kb, so d|r`
`So d is the number of conventions (B,a mod b)`
`Assuming D is the number of conventions (B,a mod b), then`
`D | B, D |r, but a = KB +r`
`So d is also the number of conventions (A, B)`
`therefore (b) and (b,a mod b) The number of conventions is the same, and their greatest common divisor are necessarily equal, to be certified`
`Euclidean algorithm is based on this principle, its algorithm is described in C + + language as:`
`int GCD (int a, int b) {if (b = = 0) return A; return Gcd (b, a% B); }`
`Of course you can also write an iterative form:`
`int GCD (int a, int b) {while (b! = 0) {int r = B; 　　 b = a% B; 　　A = R; 　　} return A; }`
`It is essentially the same principle that is used.`
`Supplement: The extended Euclidean algorithm is used to solve a set of X, Y, and A*X+B*Y=GCD (A, A, A, b) in a known A and a (A, a) (the solution must exist, according to the correlation theorem in number theory). Extended Euclidean is commonly used in solving linear equations and equations. The following is a`
`In C + + implementation:`
`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); 　　int t = x; 　　x = y; 　　 y = t-a/b * y; 　　return R; }　　`
`Comparing this implementation with the recursive implementation of GCD, we find that the following X, Y assignment process is found, which is the essence of extending Euclid's algorithm.`
`Can think like this:`
`For a ' = B, b ' = a% B, we get x, Y makes a ' x + b ' y = Gcd (a ', B ')`
`Because b ' = a% b = a-a/b * B (Note: This is a division in the programming language)`
`Then you can get:`
`A ' x + b ' y = Gcd (a ', B ') ===> bx + (a-a/b * b) y = GCD (a ', b ') = GCD (A, b) ===> ay +b (x-a/b*y) = GCD (A, B)`
`Therefore, for A and B, their relative p,q are Y and (x-a/b*y) respectively.`
`Understanding of the method of solving X, y`
`Set A>b.`
`1, obviously when B=0,GCD (A, b) =a. At this time x=1,y=0;`
`When 2,ab<>0`
`Set AX1+BY1=GCD (A, b);`
`bx2+ (a mod b) y2=gcd (b,a mod b);`
`According to the naïve Euclid principle there is gcd (A, B) =gcd (b,a MoD);`
`Then: ax1+by1=bx2+ (a mod b) y2;`
`namely: ax1+by1=bx2+ (A-(A/b) *b) y2=ay2+bx2-(A/b) *by2;`
`According to the identity theorem: x1=y2; Y1=x2-(A/b) *y2;`
`This gives us a way to solve x1,y1: The value of X1,y1 is based on X2,y2.`
`The idea above is defined recursively, because the recursive solution of GCD will always have a time b=0, so recursion can`
`End.`
`Read a lot on the internet on the problem of solving the equations of indefinite equation, can not say that all, all only said a part, after looking at a lot of real to understand the indefinite equation of the whole process, the steps are as follows:`
`An integer solution that asks A * x + b * y = c.`
`1, first calculate gcd (A, b), if n can not be divisible by gcd (A, B), then the equation has no integer solution; otherwise, the equation is divided by gcd (A, a), and the new indefinite equation a ' * x + b ' * y = C ', at this time gcd (a ', a) = 1;`
`2, using the Euclidean algorithm described above to find the equation a ' * x + b ' * y = 1 A set of integer solution X0,y0, then C ' * x0,c ' * y0 is the equation a ' * x + b ' * y = C ' A set of integer solutions;`
`3, according to the correlation theorem in number theory, can get the equation a ' * x + b ' * y = C ' of all the integer solution is:`

In fact, the solution we find is just a group,

A*X0+LCM (A, B) +B*Y0-LCM (A, b) = 1;

A*x +b*y = 1;

X=X0+B/GCD (A, b); Y=y0-a/gcd (A, b);

A/GCD (A, b) *x ' +B/GCD (A, b) *y ' =C/GCD (A, b);

X ' =c/gcd (A, b)*x0+b/gcd (A, b); y ' =c/gcd (A, b) *y0-a/GCD (A,b);

`x = C ' * x0 + b ' * t y = C ' * y0-a ' * t (t is an integer)`
`The above solution is a * x + b * y = N of all integer solutions`

Euclidean algorithm GCD and its development ultimate explanation

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