Concrete mathematics : a foundation for computer science

3.4 ‘MOD’: THE BINARY OPERATION 85
n’ = qm’ things to put into the remaining m’ = m - 1 groups. And if
r > 0, we put [n/ml = q + 1 things into the first group and replace n
by n’ = n - q - 1, leaving n’ = qm’ + T - 1 things for subsequent groups.
The new remainder is r’ = r - 1, but q stays the same. It follows that there
will be r groups with q + 1 things, followed by m - r groups with q things.
How many things are in the kth group? We’d like a formula that gives
[n/ml when k < n mod m, and Ln/m] otherwise. It’s not hard to verify
that
has the desired properties, because this reduces to q + [(r - k + 1 )/ml if we
write n = qm + r as in the preceding paragraph; here q = [n/m]. We have
[(r-k+ 1)/m] = [k