# Division Theorem/Positive Divisor/Positive Dividend/Existence/Proof 1

## Theorem

For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$:

- $\forall a, b \in \Z, a \ge 0, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$

## Proof

Let $a, b \in \Z$ such that $a \ge 0$ and $b > 0$ be given.

Let $S$ be defined as the set of all positive integers of the form $a - z b$ where $z$ is an integer:

- $S = \set {x \in \Z_{\ge 0}: \exists z \in \Z: x = a - z b}$

By setting $z = 0$ we have that $a \in S$.

Thus $S \ne \O$.

We have that $S$ is bounded below by $0$.

From Set of Integers Bounded Below by Integer has Smallest Element it follows that $S$ has a smallest element $r$.

Thus:

- $\exists q \in \Z: a - q b = r$

and so:

- $a = q b + r$

So we have proved the existence of $q$ and $r$ such that $a = q b + r$.

It remains to be shown that $0 \le r < b$.

We have that $r \in S$ which is bounded below by $0$.

Therefore $0 \le r$.

Aiming for a contradiction, suppose $b \le r$.

So:

\(\displaystyle b\) | \(\le\) | \(\displaystyle r\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle r + b\) | as $b > 0$ | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 0\) | \(\le\) | \(\displaystyle r - b\) | ||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle r\) | subtracting $b$ throughout |

But then:

\(\displaystyle a\) | \(=\) | \(\displaystyle q b + r\) | from above | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle r - b\) | \(=\) | \(\displaystyle \paren {a - q b} - b\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a - b \paren {q + 1}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle r - b\) | \(\in\) | \(\displaystyle S\) | as $r - b$ is of the correct form |

But $r - b < r$ contradicts the choice of $r$ as the least element of $S$.

Hence $r < b$ as required.

Thus the existence of $q$ and $r$ satisfying $a = q b + r, 0 \le r < b$ has been demonstrated.

$\blacksquare$

## Sources

- 1958: Martin Davis:
*Computability and Unsolvability*... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Theorem $6$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 21$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (next): $\text{A}.1$: Number theory: Theorem $\text{A}.1$