GCD : gcd igcd

### 2.6.2 GCD : `gcd igcd`

`gcd` or `igcd` denotes the gcd (greatest common divisor) of several integers (for polynomials see also 2.25.7).
`gcd` or `igcd` returns the `GCD` of all integers.
Input :

`gcd(18,15)`

Output :

`3`

Input :

`gcd(18,15,21,36)`

Output :

`3`

Input :

`gcd([18,15,21,36])`

Output :

`3`

We can also put as parameters two lists of same size (or a matrix with 2 lines), in this case `gcd` returns the greatest common divisor of the elements with same index (or of the same column). Input :

`gcd([6,10,12],[21,5,8])`

or :

`gcd([[6,10,12],[21,5,8]])`

Output :

`[3,5,4]`

An example
Find the greatest common divisor of 4n+1 and 5n+3 when n ∈ ℕ.
Input :

`f(n):=gcd(4*n+1,5*n+3)`

Then, input :

      essai(n):={
        local j,a,L;
        L:=NULL;
        for (j:=-n;j<n;j++) {
          a:=f(j);
          if (a!=1) {
            L:=L,[j,a];
          }
        }
        return L;
      }

Then, input :

`essai(20)`

Output :

`[-16,7],[-9,7],[-2,7],[5,7],[12,7],[19,7]`

So we have now to prove that :
if n!=5+k*7 (for k ∈ ℤ), 4n+1 and 5n+3 are mutually prime,
and
if n=5+k*7 (for k ∈ ℤ), the greatest common divisor of 4n+1 and 5n+3 is 7.