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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.

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

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I don't think that there is any standard term. These are the quotients in the Euclidean algorithm.

You might see terms such as "first quotient", "n-th quotient", or "final quotient", but I think that is about it.

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  • $\begingroup$ 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. $\endgroup$ – Andrey Rukhin Nov 19 '15 at 18:58

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