GCD : gcd igcd

gcd or igcd denotes the gcd (greatest common divisor) of several integers (for polynomials see also 1.25.7).
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 $\in$ $\mathbb {N}$.
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 $\in$ $\mathbb {Z}$), 4n + 1 and 5n + 3 are mutually prime,
and
if n = 5 + k*7 (for k $\in$ $\mathbb {Z}$), the greatest common divisor of 4n + 1 and 5n + 3 is 7.