How to determine if a number is even or odd with AMPScript?

I'd suggest using the MOD() AMPScript function. If @sum is evenly divisible by 2 then it's even, otherwise it's odd:

%%[

var @i, @sum, @max
set @sum = 0
set @max = 10

FOR @i = 1 TO @max DO
  SET @sum = Add(@sum,1)
NEXT @i

IF mod(@sum,2) == 0 THEN
  outputline(concat("<br>", @sum, " is even"))
ELSE
  outputline(concat("<br>", @sum, " is odd"))
ENDIF

]%%