The formal definition of the division theorem states that the quotient and remainder ( $ q, r \in \mathbb{Z} $) are said to be 'unique'.
Does that imply that if there exists quotient and remainder $q', \,r' \in \mathbb{Z} \quad then \, \, q=q' \, \, and \, \, r'=r$ has to be true?
More precisely, if $a = bq + r = bq'+r'$ where $0 \le r, r' < b$, then $q=q'$ and $r=r'$.