# Absolute Value of Integer is not less than Divisors

## Contents

## Theorem

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

- $\forall c \in \Z_{\ne 0}: a \divides c \implies a \le \size a \le \size c$

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.

### Corollary

Let $a, b \in \Z_{>0}$ be (strictly) positive integers.

Let $a \divides b$.

Then:

- $a \le b$

## Proof

Suppose $a \divides c$ for some $c \ne 0$.

From Negative of Absolute Value:

- $a \le \size a$

Then:

\(\displaystyle a\) | \(\divides\) | \(\displaystyle c\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \exists q \in \Z: \ \ \) | \(\displaystyle c\) | \(=\) | \(\displaystyle a q\) | $\quad$ Definition of Divisor of Integer | $\quad$ | |||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \size c\) | \(=\) | \(\displaystyle \size a \size q\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \size a \size q \ge \size a \times 1\) | \(=\) | \(\displaystyle \size a\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a \le \size a\) | \(\le\) | \(\displaystyle \size c\) | $\quad$ | $\quad$ |

$\blacksquare$

## Sources

- 1958: Martin Davis:
*Computability and Unsolvability*... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Corollary $2$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 11.2$: The division algorithm