Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:


Let $e\_i \in \mathbb{N} >2$, and $ r\_k \in \mathbb{N}$ where $0\le r\_k<r\_{k-1}$. Then we can do the following procedure (analogous to the standard Euclidean algorithm):
$$ \begin{array}{rcl} 
a &= & e\_1b -r\_1\\
b &= & e\_2r\_1 - r\_2\\
r\_1 &= &e\_3r\_2 -r\_3\\
&...\\
r\_{k-2} &= &e\_ir\_{k-1} = e\_i
\end{array}$$
That last line is true because the $\gcd(a,b) =1$, by hypothesis.


Considering that the greatest common divisor is known by hypothesis, what is the utility of this modified Euclidean algorithm, and what was the motivation for its development? I have heard that it has applications to toric geometry, but I am not familiar with those. 



Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
I think the motivation for the Hirzebruch-Jung algorithm is not the algorithm itself, but the fact the it yields *directly* a very interesting continued fraction expansion. It is those, now called **Hirzebruch-Jung continued fractions**, that have a wide number of applications. A bit more on that later.


Using your notation (but changing $b$ with $r\_o$ for clarity).


$$ \begin{array}{rcl} 
a &= & e\_1r\_0 -r\_1\\
r\_0 &= & e\_2r\_1 - r\_2\\
r\_1 &= &e\_3r\_2 -r\_3\\
&...\\
r\_{k-2} &= &e\_ir\_{k-1} = e\_i
\end{array}$$


By simple division you get:


$$ \begin{array}{rcl} 
a/r\_0 &= & e\_1 -r\_1/r\_0\\
r\_0/r\_1 &= & e\_2 - r\_2/r\_1\\
r\_1/r\_2 &= &e\_3 -r\_3/r\_2\\
&...\\
r\_{k-2}/r\_{k-2} &= &e\_i
\end{array}$$


And from that you can simply read off:


$$a/r\_0 = e\_1 -\frac{1}{e\_1-\frac{1}{...-\frac{1}{e\_i}}}$$


It is those finite continued fractions $a/r\_0$ that have, among many others, applications to toric varieties. To quote from D. I. Dais' "[Geometric Combinatorics in the Study of Compact Toric Surfaces](http://www.math.uoc.gr:1080/Members/ddais/Dais_Anogia.pdf)" (2000):



> 
> "Examining two-dimensional toric singularities "under the microscope"
>  one discovers a peculiar algebro-geometric world endowed with a rich
>  combinatorial structure. Viewed historically, everything begins with
>  Hirzebruch-Jung continued fractions."
> 
> 
> 


For some more motivation and applications, I quote form a very recent (2015) [paper](https://www.google.es/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCEQFjAAahUKEwjkoLfzrcbIAhUJVhoKHYVtAi8&url=http%3A%2F%2Fwww.scirp.org%2FJournal%2FPaperDownload.aspx%3FpaperID%3D59458&usg=AFQjCNGKzcZ-DFePRQjWKD7f_lFHlSdttg&sig2=K9IC1p_we5RXn9RnNPzKhg):



> 
> "Hirzebruch-Jung (H-J) continued fractions are widely used in various
>  branches of mathematics as well as in theoretical physics. First of
>  all, HJ-continued fractions arise naturally in the minimal resolution
>  of cyclic quotient (that is, Hirzebruch-Jung) surface singularities of
>  the type $\mathbb{C}/\mathbb{Z}\_p$, which is also known as
>  HJ-resolution" [...]
> 
> 
> HJ-continued fractions are used also to describe the plumbing
>  decomposition of the other type of link of surface singularities,
>  namely, Seifert fibered homology spheres (Sfh-spheres), particularly
>  Brieskorn homology spheres (Bh-spheres) [...]
> 
> 
> In condensed matter theory, the HJ-continued fractions are used to
>  describe fractional quantum Hall (FQH) systems with k levels of
>  hierarchy [...]
> 
> 
> Also, in topological string theory, the structure of internal space
>  (Calabi-Yau threefold) can be encoded in terms of HJ-continued
>  fraction expansion of positive integers, which are the topological
>  invariants of the internal space and define the mode of interactions
>  between D-branes"
> 
> 
> 


