If $a$ and $b$ are integers, with $b \not = 0$, then there exist unique integers $q$ and $r$ such that $a= b \cdot q + r$ where $0 \leq r <|b|$.

 Corollary: If $a$ and $b$ are integers, with $b \not = 0$, then there exist unique integers $q$ and $r$ such that $a= b \cdot q + r$ where  $0 \leq  r <|b|$.

Proof: Consider the case in which $b$ is negative. So $|b| > 0$. By  Division Algorithm Theorem, there exist unique integers $q'$ and $r$ such that \linebreak $a=  |b| \cdot  q'+ r$ where  $0 \leq  r <|b|$. Note that $|b | = - b$, we can take $q = -q'$ to get $a= q \cdot b + r$,  where  $0 \leq  r <|b|$. Consider the case in which $b$ is positive. Proof follows from Division Algorithm Theorem.

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