# Standard terminology for the “quotient of a quotient”

Let $n$ and $m$ be integers. One can write $$n = mq_0 + r_0$$ where $0\leq r_0 < m$; the $q_0$ term in the right-hand side is the "quotient". One could then write $$q_0 = mq_1 + r_1$$ where $0\leq r_1 < m$. What is the standard terminology for the $q_1$ term (the "quotient of the quotient")? Proceeding inductively, if we define $$q_i = mq_{i+1} + r_i$$ where $0\leq r_i < m$ for all $i\geq 0$, then what is the terminology for the terms $q_0,q_1,q_2,\ldots,$?

I am $not$ applying the Euclidean algorithm here.

• I'm afraid you're making some mistakes in applying the Euclidean algorithm, see here: en.wikipedia.org/wiki/Euclidean_algorithm#Procedure – PITTALUGA Nov 19 '15 at 19:07
• I am not applying the Euclidean algorithm (see my comment below). – Andrey Rukhin Nov 19 '15 at 20:49
• so $q_k =n \mod m^{k+1}$ – miracle173 Nov 20 '15 at 8:42

It's the algorithm to write the digits of the number n in the numbersystem with base m I think. The $r_k$ are then the digits, but I do not know a name for the $q_k$. $q_0 = \lfloor n / m \rfloor$ ,$q_1 = \lfloor q_0 / m \rfloor$ ...
Example: if $m=10$ One could ask: if $n=1234321$ what is the name for the part $q_0=123432$ ? or for the part $q_2=1234$?
It's not exactly "the leading digits", the best I could name it were " $q_k$ is the floor-function of $n / m^{k+1}$ "
• These quotients do not align with the Euclidean algorithm. When computing the $\gcd$ of $m$ and $n$, one doesn't compute the quotient with respect to $m$ at each iteration. Thanks for the response. – Andrey Rukhin Nov 19 '15 at 18:58