Definition:Proper Divisor/Integer

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\struct {\Z, +, \times}$ be the ring of integers.

Let $x, y \in \Z$.


Then $x$ divides $y$ is defined as:

$x \divides y \iff \exists t \in \Z: y = t \times x$


Then $x$ is a proper divisor of $y$ if and only if:

$(1): \quad x \divides y$
$(2): \quad \size x \ne \size y$
$(3): \quad x \ne \pm 1$

That is:

$(1): \quad x$ is a divisor of $y$
$(2): \quad x$ and $y$ are not equal in absolute value
$(3): \quad x$ is not equal to either $1$ or $-1$.


Also known as

A proper divisor is also known as a proper factor.


Also see


Sources