Theorem. Let $a,\,b$ be non-zero elements of a Euclidean domain $D$ with the Euclidean valuation $\nu$ . The incomplete quotient $q$ and the remainder $r$ of the division algorithm $$a = qb+r \quad \mbox{where} \quad r = 0 \;\; \mbox{or}\;\; \nu(r) < \nu(b)$$ are unique if and only if

(1)

Conversely, assume that (1) is not true for non-zero elements $a,\,b$ of $D$ , i.e. $$\nu(a+b) > \max\{\nu(a),\,\nu(b)\}.$$ Then we obtain two repsesentations $$b = 0(a+b)+b = 1(a+b)-a$$ where $\nu(b) < \nu(a+b)$ and $\nu(-a) = \nu(a) < \nu(a+b)$ . Thus the incomplete quotient and the remainder are not unique.