Suppose that a is an integer. If a is nonzero, not every integer b is a divisor of a. If b (not equal to 0) does not divide a, then there is a remainder after division of a by b. Here is a precise statement about division with remainder.
Theorem
If
a 
Z and b
N\{0}, then there are exactly one q
Z and one r
Z such that
a = qb + r and 0
r < b.
This theorem states that there exist a quotient q and a
remainder r, but it does not tell you how to find those two
integers. However, the
proof that we give here is constructive: it provides an algorithm
to find q and r.
Notation: The remainder r is often denoted
by a mod b.