Greatest common divisor, least common multiple "number theory"

Function: Greatest common divisor of A and B

Incoming parameters: integer A, integer b

Outgoing parameters: Greatest common divisor of A and B

Algorithm 1: Euclidean algorithm

Time complexity: O (N)

Implementation principle:

For A, B (a>b) of two, the steps for A and B greatest common divisor (b) are as follows:

B In addition to a, get A÷b=q......r1 (0≤R1). If r1=0, then (A, b) =b;

If r1≠0, then use R1 except B, get B÷r1=q......r2 (0≤R2).

If r2=0, then (A, B) =r1, if r2≠0, then continue to use R2 in addition to R1, ...

So go on until you can divide it evenly. The last non-0 divisor is (A, b).

Implementation code:

#include <iostream> #include <algorithm> #include <cstdio> #include <cstring> using namespace Std;template <typename t>t GCD (t a,t b) {    if (a < b)    {        T temp = A;        A = b;        b = temp;    }    if (b = = 0)        return A;    Return GCD (b,a%b);} int main () {    int a = 5,b = 6;    printf ("%d\n", GCD (A, b));    return 0;}

Algorithm 2:stein algorithm

Time complexity: O (log (max (b)))

Implementation principle:

GCD (A,a) =a, that is, a number and its own number of conventions is still its own.
GCD (KA,KB) =k gcd (A, b), that is, the greatest common divisor operation and the multiply multiplication operation can be exchanged.

In particular, when k=2, two even-numbered greatest common divisor must be divisible by 2.
When K and B are mutually prime, gcd (ka,b) =GCD (A, b), that is, about two digits, only one of them

A contained factor does not affect the greatest common divisor. Specifically, when k=2, the description calculates a

When even and an odd greatest common divisor, you can divide the even number by 2.

Algorithm process:

1, if an=bn, then an (or Bn) *CN is greatest common divisor, the algorithm ends
2. If an=0,bn is greatest common divisor, the algorithm ends
3. If Bn=0,an is greatest common divisor, the algorithm ends
4. Set A1=a, B1=b and C1=1
5, if an and Bn are even, then an+1=an/2,bn+1=bn/2,cn+1=cn*2 (note, multiply 2 to move the integer to the left one can, except 2 to move the integer right one can)
6, if an is even, Bn is not even, then AN+1=AN/2,BN+1=BN,CN+1=CN (obviously, 2 is not an odd number of approximate)
7, if the Bn is even, an is not even, then BN+1=BN/2,AN+1=AN,CN+1=CN (obviously, 2 is not an odd number of approximate)
8, if an and bn are not even, then an+1=| An-bn|/2,bn+1=min (an,bn), CN+1=CN
9, N plus 1, ext. 1

Implementation code:

#include <iostream> #include <algorithm> #include <cstdio> #include <cstring>using namespace Std;template <typename t>t GCD (T a,t b) {    if (a = = 0) return b;    if (b = = 0) return A;    if (~a & 1)    {        if (b & 1)            return GCD (a >> 1, b);        else            return GCD (a >> 1, b >> 1) << 1;    }    if (~b & 1)        return GCD (A, B >> 1);    if (a > B)        return GCD ((a) >> 1, b);    

Max Common multiple:

Implementation code:

Template <typename t>t LCM (T a,t b) {return a/gcd (a b) *b;}

Greatest common divisor, least common multiple "number theory"