# 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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