[03] Divisibility in the Set of Integers Z

Division Algorithm in Z

Given the integers a and b, with b > 0, there exists unique integers

q and r satisfying

a = bq + r such that 0 ≤ r < b.

The unique integers q and r are called, respectively, the quotient

and remainder in the division of a by b.

even

If b = 2, then either r = 0 or 1. When r = 0, then a = 2q

and the integer a is said to be

odd

if a = 2q +1, then a is said to be

divisible

An integer b is said to be ________ by a nonzero integer a, in symbols a | b if there exists some integer k such that b = ak. We write a”| b to indicate that b is not ______ by a.