Division Theorem/Positive Divisor/Positive Dividend
Jump to navigation
Jump to search
Theorem
For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \le r < b$:
- $\forall a, b \in \Z, a \ge 0, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
In the above equation:
- $a$ is the dividend
- $b$ is the divisor
- $q$ is the quotient
- $r$ is the principal remainder, or, more usually, just the remainder.
Proof
This result can be split into two parts:
Proof of Existence
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 of Uniqueness
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:
- $\forall a, b \in \Z, a \ge 0, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$