Example: Divisibility Examples

(related to Definition: Divisor, Complementary Divisor, Multiple)

Example 1

We provide a list of all divisors of $24:$

$$\begin{array}{c} \pm 1\mid 24\\ \pm 2\mid 24\\ \pm 3\mid 24\\ \pm 4\mid 24\\ \pm 6\mid 24\\ \pm 8\mid 24\\ \pm 12\mid 24\\ \pm 24\mid 24\\ \end{array}$$

Example 2

Divisors and complementary divisors of $81:$

$$\begin{array}{c} \pm 1\mid 81\\ \pm 3\mid 81\\ \pm 9\mid 81\\ \pm 27\mid 81\\ \pm 81\mid 81\\ \end{array}$$

Example 3

Divisors and complementary divisors of $101:$

$$\begin{array}{c} \pm 1\mid 101\\ \pm 101\mid 101\\ \end{array}$$

These examples demonstrate that the number of divisors of an integer is not constant. Some integers seem to have more divisors than others. Another observation is that the sign of the divisors seems not to play any role in the divisibility relation. This observation will be formulated and proven in the following proposition.

Thank you to the contributors under CC BY-SA 4.0!




  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013