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:
Given integers a and b, with , there exist unique integers q and r satisfying
The integers q and r, are called, respectively, the quotient and remainder in the division of a by b.
We propose to show that the expression is an integer for all . According to the Division Algorithm, every a is of the form 3q, , or . Assume the first of these cases. Then,
, which clearly is an integer. Similarly, if , then
and is an integer in this instance also. Finally, for , we obtain
an integer once more. Consequently, our result is established in all cases.