Integer Absolute Value Greater than Divisors

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Theorem

A (non-zero) integer is greater than or equal to its divisors in magnitude:

$\forall c \in \Z^*: a \backslash c \implies a \le \left\vert{a}\right\vert \le \left\vert{c}\right\vert$

It follows that a non-zero integer can have only a finite number of divisors, since they must all be less than or equal to it.

Let $a, b \in \Z$.

If $a$ and $b$ are both positive, and $a \backslash b$, then $a \le b$.

Suppose $a \backslash c, c \ne 0$. It's a given that $a \le \left\vert{a}\right\vert$.

$\displaystyle $	$\displaystyle a \backslash c$	$\implies$	$\displaystyle \exists q \in \Z: c = a q$	$\displaystyle $	Definition of divisor
$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left\vert{c}\right\vert = \left\vert{a}\right\vert \left\vert{q}\right\vert$	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left\vert{a}\right\vert \left\vert{q}\right\vert \ge \left\vert{a}\right\vert \times 1 = \left\vert{a}\right\vert$	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle a \le \left\vert{a}\right\vert \le \left\vert{c}\right\vert$	$\displaystyle $

$\blacksquare$

Follows directly.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle a \backslash c\)	\(\implies\)	\(\displaystyle \exists q \in \Z: c = a q\)	\(\displaystyle \)	Definition of divisor
\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left\vert{c}\right\vert = \left\vert{a}\right\vert \left\vert{q}\right\vert\)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left\vert{a}\right\vert \left\vert{q}\right\vert \ge \left\vert{a}\right\vert \times 1 = \left\vert{a}\right\vert\)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle a \le \left\vert{a}\right\vert \le \left\vert{c}\right\vert\)	\(\displaystyle \)