See also divmod/4 (not visible in the function in the header list because it is listed under "special arithmetic functions")

A little exercise to compare the pairs

div,mod

and

//,rem

can be found here

With output:

?- test.
( -6 mod 3) =  0, ( -6 div 3) = -2, OK : 3*-2+0=-6
( -5 mod 3) =  1, ( -5 div 3) = -2, OK : 3*-2+1=-5
( -4 mod 3) =  2, ( -4 div 3) = -2, OK : 3*-2+2=-4
( -3 mod 3) =  0, ( -3 div 3) = -1, OK : 3*-1+0=-3
( -2 mod 3) =  1, ( -2 div 3) = -1, OK : 3*-1+1=-2
( -1 mod 3) =  2, ( -1 div 3) = -1, OK : 3*-1+2=-1
(  0 mod 3) =  0, (  0 div 3) =  0, OK : 3*0+0=0
(  1 mod 3) =  1, (  1 div 3) =  0, OK : 3*0+1=1
(  2 mod 3) =  2, (  2 div 3) =  0, OK : 3*0+2=2
(  3 mod 3) =  0, (  3 div 3) =  1, OK : 3*1+0=3
(  4 mod 3) =  1, (  4 div 3) =  1, OK : 3*1+1=4
(  5 mod 3) =  2, (  5 div 3) =  1, OK : 3*1+2=5
(  6 mod 3) =  0, (  6 div 3) =  2, OK : 3*2+0=6

( -6 mod -3) =  0, ( -6 div -3) =  2, OK : -3*2+0=-6
( -5 mod -3) = -2, ( -5 div -3) =  1, OK : -3*1+-2=-5
( -4 mod -3) = -1, ( -4 div -3) =  1, OK : -3*1+-1=-4
( -3 mod -3) =  0, ( -3 div -3) =  1, OK : -3*1+0=-3
( -2 mod -3) = -2, ( -2 div -3) =  0, OK : -3*0+-2=-2
( -1 mod -3) = -1, ( -1 div -3) =  0, OK : -3*0+-1=-1
(  0 mod -3) =  0, (  0 div -3) =  0, OK : -3*0+0=0
(  1 mod -3) = -2, (  1 div -3) = -1, OK : -3*-1+-2=1
(  2 mod -3) = -1, (  2 div -3) = -1, OK : -3*-1+-1=2
(  3 mod -3) =  0, (  3 div -3) = -1, OK : -3*-1+0=3
(  4 mod -3) = -2, (  4 div -3) = -2, OK : -3*-2+-2=4
(  5 mod -3) = -1, (  5 div -3) = -2, OK : -3*-2+-1=5
(  6 mod -3) =  0, (  6 div -3) = -2, OK : -3*-2+0=6

Below with current_prolog_flag(integer_rounding_function,toward_zero)

( -6 rem 3) =  0, ( -6 // 3) = -2, OK : 3*-2+0=-6
( -5 rem 3) = -2, ( -5 // 3) = -1, OK : 3*-1+-2=-5
( -4 rem 3) = -1, ( -4 // 3) = -1, OK : 3*-1+-1=-4
( -3 rem 3) =  0, ( -3 // 3) = -1, OK : 3*-1+0=-3
( -2 rem 3) = -2, ( -2 // 3) =  0, OK : 3*0+-2=-2
( -1 rem 3) = -1, ( -1 // 3) =  0, OK : 3*0+-1=-1
(  0 rem 3) =  0, (  0 // 3) =  0, OK : 3*0+0=0
(  1 rem 3) =  1, (  1 // 3) =  0, OK : 3*0+1=1
(  2 rem 3) =  2, (  2 // 3) =  0, OK : 3*0+2=2
(  3 rem 3) =  0, (  3 // 3) =  1, OK : 3*1+0=3
(  4 rem 3) =  1, (  4 // 3) =  1, OK : 3*1+1=4
(  5 rem 3) =  2, (  5 // 3) =  1, OK : 3*1+2=5
(  6 rem 3) =  0, (  6 // 3) =  2, OK : 3*2+0=6

( -6 rem -3) =  0, ( -6 // -3) =  2, OK : -3*2+0=-6
( -5 rem -3) = -2, ( -5 // -3) =  1, OK : -3*1+-2=-5
( -4 rem -3) = -1, ( -4 // -3) =  1, OK : -3*1+-1=-4
( -3 rem -3) =  0, ( -3 // -3) =  1, OK : -3*1+0=-3
( -2 rem -3) = -2, ( -2 // -3) =  0, OK : -3*0+-2=-2
( -1 rem -3) = -1, ( -1 // -3) =  0, OK : -3*0+-1=-1
(  0 rem -3) =  0, (  0 // -3) =  0, OK : -3*0+0=0
(  1 rem -3) =  1, (  1 // -3) =  0, OK : -3*0+1=1
(  2 rem -3) =  2, (  2 // -3) =  0, OK : -3*0+2=2
(  3 rem -3) =  0, (  3 // -3) = -1, OK : -3*-1+0=3
(  4 rem -3) =  1, (  4 // -3) = -1, OK : -3*-1+1=4
(  5 rem -3) =  2, (  5 // -3) = -1, OK : -3*-1+2=5
(  6 rem -3) =  0, (  6 // -3) = -2, OK : -3*-2+0=6
true.