The Division Algorithm

One theorem, the Division Algorithm, acts as the foundation stone upon which our whole development of Number Theory rests. The result is familiar to most of us; roughly, it asserts that an integer a can be “divided” by a positive integer b in such a way that the remainder is smaller than b. The exact statement is given below:

Division Algorithm.

Given integers a and b, with $b>0$ , there exist unique integers q and r satisfying

$a=qb+r$ , $0 \leq r < b$ .

The integers q and r, are called, respectively, the quotient and remainder in the division of a by b.

Example.

We propose to show that the expression $a(a^{2}+2)/3$ is an integer for all $a \geq 1$ . According to the Division Algorithm, every a is of the form 3q, $3q+1$ , or $3q+2$ . Assume the first of these cases. Then,