Looking at division and mod as a pair finding group and subgroup indices is interesting. For zero based indexing, we want division to round down and the usual

Looking at division and mod as a pair finding group and subgroup indices is interesting. For zero based indexing, we want division to round down and the usual zero-based mod. So Group(8,4) = 8/4 = 2, Index(8,4) = 8 % 4 = 0. The invariant is x == x/y * y + x % y. With one based indexing, we want division to round up and want the modulo operator you're describing. So Group(8,4) = 8/'4 = 2, Index(8,4) = 8 %' 4 = 4. The invariant is x == (x/'y - 1) * y + x %' y.