Alignment of multiple equations and their comments

Question

I need to center-align a series of equations and then align their comments (which are themselves to be right-justified (except not all of them!). I'm slightly overwhelmed by the numerous options available for alignment of equations and can't seem to find an appropriate example (online or in the docs). Thanks for your help/suggestions in advance.

\documentclass[12pt, leqno]{book}
\usepackage{mathtools,amsthm}

\begin{document}
\noindent 2-parter: First, eqns. (1) and (2) should be in alignment (centered on page). 
 Here, (1) is pushed/pulled left while (2) is perfect. Secondly, the last comment 
 ($0 < r_{k} < r_{k-1},$) of (1) should be right-justified as is the comment of (2), with 
 the final form then being all ``0"s in alignment.
 \begin{flalign}
    \begin{aligned}
             a &= qb + r,               &0 < r < b,                   \\
             b &= q_{1}r + r_{1},       &0 < r_{1} < r,               \\
             r &= q_{2}r_{1} + r_{2},   &0 < r_{2} < r_{1},           \\
         r_{1} &= q_{3}r_{2} + r_{3},   &0 < r_{3} < r_{2},           \\
        &\texttt{.\quad.\quad.\quad.}   &\texttt{.\quad.\quad.\quad.} \\
       r_{k-2} &= q_{k}r_{k-1} + r_{k}, &0 < r_{k} < r_{k-1},         \\
       r_{k-1} &= q_{k+1}r_{k}.
    \end{aligned}
 \end{flalign}
 %% Thanks, @Bernard for your help with the following eqn
 \begin{flalign}
    && a = qb + r, && \llap{$0 \leq r < b$,} 
 \end{flalign}
 \end{document}

Answer 1

Here is a way to do it: I defined an \eqmathbox command, derived from the \eqmakebox command. manually, I could align the last equation. I also replaced the horizontal dots with \vdotswithin{=} from math tools:

\documentclass[12pt, leqno]{book}
\usepackage{mathtools,amsthm}
\usepackage{eqparbox}
\newcommand\eqmathbox[2][]{\llap{\eqmakebox[#1][l]{\ensuremath{\displaystyle#2}}}}


\begin{document}

\noindent 2-parter: First, eqns. (1) and (2) should be in alignment (centered on page).
Here, (1) is pushed/pulled left while (2) is perfect. Secondly, the last comment
($0 < r_{k} < r_{k-1},$) of (1) should be right-justified as is the comment of (2), with
the final form then being all ``0"s in alignment.

\begin{flalign}
  \notag & & a &= qb + r, & & \eqmathbox[C]{0 < r < b,}\\
 \notag & & b &= q_{1}r + r_{1}, & & \eqmathbox[C]{0 < r_{1} < r,} \\
 \notag & & r &= q_{2}r_{1} + r_{2}, & & \eqmathbox[C]{0 < r_{2} < r_{1},} \\
  & & r_{1} &= q_{3}r_{2} + r_{3}, & & \eqmathbox[C]{0 < r_{3} < r_{2},} \\
  \notag & & &\vdotswithin{=} & & \eqmathbox[C]{\hspace*{2em}\vdots}\\
 \notag & & r_{k-2} &= q_{k}r_{k-1} + r_{k}, & & \eqmathbox[C]{0 < r_{k} < r_{k-1},} \\
 \notag & & r_{k-1} &= q_{k+1}r_{k}.
\end{flalign}

\begin{flalign}
  && a & = qb + r,\quad &&\mathllap{0 \leq r < b,}
\end{flalign}

\end{document} 

Answer 2

The following seems to achieve what you're after:

\documentclass[leqno]{article}
\usepackage{mathtools}

\newlength{\mathlen}
\begin{document}

\settowidth{\mathlen}{$0 < r_k < r_{k-1},$}% Widest comment on the right
\newcommand{\rightcomment}[1]{\makebox[0pt][r]{\makebox[\mathlen][l]{$#1$}}}% Simplify code with a macro
\begin{flalign}
  &&          a &= qb + r,            &\rightcomment{0 < r < b,}                         \nonumber \\
  &&          b &= q_1 r + r_1,       &\rightcomment{0 < r_1 < r,}                       \nonumber \\
  &&          r &= q_2 r_1 + r_2,     &\rightcomment{0 < r_2 < r_1,}                     \nonumber \\
  &&        r_1 &= q_3 r_2 + r_3,     &\rightcomment{0 < r_3 < r_2,}                               \\
  &&            &\vdotswithin{=}      &\rightcomment{\phantom{0 < {}} \vdotswithin{r_k}} \nonumber \\
  &&    r_{k-2} &= q_k r_{k-1} + r_k, &\rightcomment{0 < r_k < r_{k-1},}                 \nonumber \\
  &&    r_{k-1} &= q_{k+1} r_k.                                                          \nonumber
\end{flalign}

For reference
\begin{equation}
  \mathmakebox[0pt]{
    \mathmakebox[\linewidth]{
      \phantom{r_{k-2}}\mathllap{a} = \mathrlap{qb + r}\phantom{q_{k}r_{k-1} + r_k,}
    }
    \rightcomment{0 \leq r < b,}
  }
\end{equation}

\end{document}

The vertical rules denote the text block boundary as indicated by showframe.

Answer 3

If you want to align the second column, you should add an extra &. The odd & characters are used to vertically align your equations and the even ones create a separation (fill) between the columns.

I can't get the flalign to work on my computer, so here a quick example with align. It shouldn't be heard to implement this in your document as well.

\begin{align*}
             a &= qb + r,             & & 0 < r < b,                   \\
             b &= q_{1}r + r_{1},     & & 0 < r_{1} < r,               \\
             r &= q_{2}r_{1} + r_{2}, & & 0 < r_{2} < r_{1},           \\
         r_{1} &= q_{3}r_{2} + r_{3}, & & 0 < r_{3} < r_{2},           \\
        &\texttt{.\quad.\quad.\quad.} & & \texttt{.\quad.\quad.\quad.} \\
       r_{k-2} &= q_{k}r_{k-1} + r_{k},& &0 < r_{k} < r_{k-1},         \\
       r_{k-1} &= q_{k+1}r_{k}. \\[.6cm]
               & a = qb + r, && 0 \leq r < b
    \end{align*}
 \end{document}