Division Theorem/Positive Divisor/Existence/Proof 2

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Theorem

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

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


Proof

Let:

$q = \floor {\dfrac a b}, t = \dfrac a b - \floor {\dfrac a b}$

where $\floor {\, \cdot \,}$ denotes the floor function.

Thus $q \in \Z$ and $t \in \hointr 0 1$.

So:

$\dfrac a b = q + t$

and so:

$(1): \quad a = q b + r$

where $r = t d$.

Since $a, q, b \in \Z$, it follows from $(1)$ that:

$r = a - q b$

and so $r \in \Z$ also.

Since $0 \le t < 1$ and $b > 0$, it follows that:

$0 \le t b < b$

that is:

$0 \le r < b$

$\blacksquare$


Sources

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