Division Theorem/Positive Divisor/Positive Dividend/Uniqueness

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Theorem

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

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


Proof 1

Suppose $q_1, r_1$ and $q_2, r_2$ are two pairs of $q, r$ that satisfy $a = q b + r, 0 \le r < b$.

That is:

\(\displaystyle a\) \(=\) \(\displaystyle q_1 b + r_1, 0 \le r_1 < b\)
\(\displaystyle a\) \(=\) \(\displaystyle q_2 b + r_2, 0 \le r_2 < b\)


This gives:

$0 = b \paren {q_1 - q_2} + \paren {r_1 - r_2}$


Aiming for a contradiction, suppose that $q_1 \ne q_2$.

Without loss of generality, suppose that $q_1 > q_2$.

Then:

\(\displaystyle q_1 - q_2\) \(\ge\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle r_2 - r_1\) \(=\) \(\displaystyle b \left({q_1 - q_2}\right)\)
\(\displaystyle \) \(\ge\) \(\displaystyle b \times 1\) as $b > 0$
\(\displaystyle \) \(=\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle r_2\) \(\ge\) \(\displaystyle r_1 + b\)
\(\displaystyle \) \(\ge\) \(\displaystyle b\)

This contradicts the assumption that $r_2 < b$.

A similar contradiction follows from the assumption that $q_1 < q_2$.

Therefore $q_1 = q_2$ and so $r_1 = r_2$.

Thus it follows that $q$ and $r$ are unique.

$\blacksquare$


Proof 2

It is given by Division Theorem: Positive Divisor: Positive Dividend: Existence that such $q$ and $r$ exist.


Let $a = 0$.

It is immediately apparent that $q = r = 0$ is the only possible solution with $0 \le r < b$.

$\Box$


Let $a > 0$ and $b = 1$.

Then from the condition $0 \le r < b$ it follows that $r = 0$.

Hence $r = 0, q = a$ is the only possible solution.

$\Box$


Let $a > 0$ and $b > 1$.

By the Basis Representation Theorem, $a$ has a unique representation to the base $b$:

\(\displaystyle a\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^s r_k b^k\)
\(\displaystyle \) \(=\) \(\displaystyle b \sum_{k \mathop = 0}^{s - 1} r_k b^{k - 1} + r_0\)
\(\displaystyle \) \(=\) \(\displaystyle b q + r\) where $0 \le r = r_0 < b$


Suppose a second pair $q', r'$ were to exist.

Then there would be a representation for $q'$ to the base $b$:

$\displaystyle q' = \sum_{k \mathop = 0}^t u_k b^k$

so that:

\(\displaystyle a\) \(=\) \(\displaystyle b q' + r'\) $0 \le r < b$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^t u_k b^{k + 1} + r'\)

But there already exists a representation of $a$ to the base $b$:

$\displaystyle a = \sum_{k \mathop = 0}^s r_k b^k$

By the Basis Representation Theorem, such a representation is unique.

So:

\(\displaystyle t\) \(=\) \(\displaystyle s - 1\)
\(\displaystyle u_k\) \(=\) \(\displaystyle r_{k + 1}\)
\(\displaystyle r'\) \(=\) \(\displaystyle a_0\)
\(\displaystyle \) \(=\) \(\displaystyle r\)

and hence $q' = q$.

$\blacksquare$