# Based on the use of Euclidean algorithm

- 2020-04-01 21:40:30
- OfStack

Euclidean algorithm is called division, used to find the common factor of two known natural Numbers m and n. Combine the procedure to explain the specific case of division.

**
First, the recursive implementation:
**

```
int getcd(int m,int n)
{
if (m < 0 || n <0) {
return 0;
}
if(m < n)
{
int t = m;
m = n;
n = t;
}
if(m % n)
{
return getcd(n,(m % n));
}
else
{
return n;
}
}
```

**The main calculation process is divided into three steps:**

1. The two input natural Numbers m > N has a remainder r such that 0< = r < n

2. If r is 0, n is the result and returns directly

3. If r is not 0, then m=n is assigned, and n=r is reexecuted from step 1

The definition of the common factor of two natural Numbers explains the conditions under which the calculated results are produced. If the remainder calculated in step 1 r = 0, the smaller number is a common factor. If r! =0 then the relation between the natural Numbers m and n can be expressed as: m = kn + r (where k is the natural number). The equation can prove that any number divisible into m must be divisible into n and r. The equation can be further deformed as: r = m-kn, which means that any number divisible into m and n must also be divisible into r. In other words, the set of Numbers that are divisible into m and n is the same as the set of Numbers that are divisible into n and r. So the division method works.

Release another version of the loop that implements Euclid's algorithm.

```
int getcd2(int m,int n)
{
if (m < 0 || n <0) {
return 0;
}
if(m<n)
{
int t=m;
m=n;
n=t;
}
int cd = 1;
while(1){
int r = m % n;
if(0==r)
{
cd = n;
break;
}
else {
m=n;
n=r;
}
}
return cd;
}
```