Question: Consider the following code:

```
\documentclass{article}

\begin{document}

\noindent
Can I make \LaTeX{} reduce a fraction automatically?\\[\baselineskip]
For example, I would like the fraction
\[ \frac{278\,922}{74\,088} \]
to be reduced to
\[ \frac{6641}{1764} \]

\end{document}

```

![output](https://i.stack.imgur.com/wUiA2.jpg)

P.S. The numerator and denominator are always both natural numbers in my case.

Answer: 
Here is a solution using **R** for the computations and the R-package `knitr` to link back to the LaTeX file. 

```
\documentclass{article}

\begin{document}
Using R with the package 'knitr' to reduce the fraction and then get both the reduced fraction but also the components of the fraction. 

Note: This is a quick demo of linking R and LaTeX using the 'knitr' package. This has been run on Windows 8.1, with MikTeX 2.9, and TeXmaker 4.4.1 as the IDE. The following code is saved as \textbf{knit02.Rnw} (and this is case sensitive). With the package 'knitr' installed in R 3.1.3 you run the command: \emph{knit("knit02.Rnw")}. This will generate the file \textbf{knit02.tex} which you now compile with pdflatex and view as a pdf.

<<echo=FALSE>>=
library(MASS)
## This function is from http://stackoverflow.com/questions
## /14820029/getting-numerator-and-denominator-of-a-fraction-in-r
getfracs <- function(frac) {
  tmp <- strsplit(attr(frac,"fracs"), "/")[[1]]
  list(numerator=as.numeric(tmp[1]),denominator=as.numeric(tmp[2]))
}
dd<-278922
nn<-74088
x<- fractions(dd/nn)
fracs<-getfracs(x)
denom<-fracs$denominator
numer<-fracs$numerator
@
\medskip

The original fraction is $\displaystyle{\frac{\Sexpr{as.integer(dd)}}{\Sexpr{as.integer(nn)}}}$.
\medskip
The reduced fraction components are \Sexpr{fracs}.
\medskip

The reduced denominator is \Sexpr{denom}.
\medskip

The reduced numerator is \Sexpr{numer}.
\medskip

And the reduced fraction is $\displaystyle{\frac{\Sexpr{denom}}{\Sexpr{numer}}}$

\end{document}

```

And the output using `pdflatex`:

![enter image description here](https://i.stack.imgur.com/U8GiF.png)
There's already another LuaTeX answer, but IMO it doesn't handle properly with locality and integer division. So here, and taking advantage of the fact Lua 5.3+ [includes bitwise operators](https://www.lua.org/manual/5.3/manual.html#3.4.2), a different approach using a [binary GCD algorithm](https://xlinux.nist.gov/dads/HTML/binaryGCD.html):

```
\documentclass{standalone}
%\usepackage{amsmath}
\usepackage{luacode}
\begin{luacode*}
userdata = userdata or {}

--https://xlinux.nist.gov/dads/HTML/binaryGCD.html
userdata.gcd = function (u,v)
    --To handle with negative values
    local u, v, g = math.abs(u), math.abs(v), 1
    --Nice error message
    assert(v~=0, "Denominator cannot be zero")
    while u&1==0 and v&1==0 do
        u=u>>1
        v=v>>1
        g=g<<1
    end
    while u>0 do
        if u&1==0 then 
            u=u>>1
        elseif v&1==0 then
            v=v>>1
        else 
            local t = math.abs(u-v)>>1
            if u<v then v=t else u=t end
        end
    end
    return v*g
end

userdata.simplified = function(u,v)
    local gcd = userdata.gcd(u,v)
    tex.sprint(("\\frac{%d}{%d}")
    :format(u//gcd, v//gcd))
end

\end{luacode*}
\newcommand\simplified[2]{\directlua{userdata.simplified(#1,#2)}}
\begin{document}
$\displaystyle\simplified{278922}{74088}$
\end{document}

```

[![enter image description here](https://i.stack.imgur.com/MuxMU.png)](https://i.stack.imgur.com/MuxMU.png)

The binary algorithm is slightly faster, but that's only noticeable when simplified fractions are extensively used in a document.