  The integer division of a numerator n by a divisor d gives a quotient q and a
remainder r. Optimizing compilers accelerate software by replacing the division
of n by d with the division of c * n (or c * n + c) by m for convenient
integers c and m chosen so that they approximate the reciprocal: c/m ~= 1/d.
Such techniques are especially advantageous when m is chosen to be a power of
two and when d is a constant so that c and m can be precomputed. The literature
contains many bounds on the distance between c/m and the divisor d. Some of
these bounds are optimally tight, while others are not. Using accessible
mathematics, we present optimally tight bounds for quotient and remainder
computations.
