information:classification
Quotients

If m and n are integers and n ≠ 0,

If n is a positive integer

If m is positive

For m = 2 these imply

More generally, for positive m (See Hermite's identity)

The following can be used to convert floors to ceilings and vice versa (m positive)

For all m and n strictly positive integers:

which, for positive and coprime m and n, reduces to

and similarly for the ceiling and fractional part functions (still for positive and coprime m and n),

Since the right-hand side of the general case is symmetrical in m and n, this implies that

More generally, if m and n are positive,

This is sometimes called a reciprocity law.

Division by positive integers gives rise to an interesting and sometimes useful property. Assuming m, n > 0 {\displaystyle m,n>0},

Similarly,

Indeed,

keeping in mind that ⌊ x / n ⌋ = ⌊ ⌊ x ⌋ / n ⌋. {\textstyle \lfloor x/n\rfloor ={\bigl \lfloor }\lfloor x\rfloor /n{\bigr \rfloor }.} The second equivalence involving the ceiling function can be proved similarly.