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,

\frac{a(a^{2}+2)}{3}=q(9q^{2}+2), which clearly is an integer. Similarly, if a=3q+1, then

\frac{(3q+1)((3q+1)^{2}+2)}{3}=(3q+1)(3q^{2}+2q+1)

and a(a^{2}+2)/3 is an integer in this instance also. Finally, for a=3q+2, we obtain

\frac{(3q+2)((3q+2)^{2}+2)}{3}=(3q+2)(3q^{2}+4q+2)

an integer once more. Consequently, our result is established in all cases.

More later,

Nalin Pithwa

 

 

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