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: 
If you are not bound to [`expl3`](https://www.ctan.org/pkg/expl3) (in which case you “just” need to implement the algorithm):

```
\documentclass{scrartcl}
\usepackage{xintgcd,xintfrac}

\newcommand*\reducedfrac[2]
  {\begingroup
     \edef\gcd{\xintGCD{#1}{#2}}%
     \frac{\xintNum{\xintDiv{#1}{\gcd}}}{\xintNum{\xintDiv{#2}{\gcd}}}%
   \endgroup}

\begin{document}
\[
  \frac{278922}{74088} = \reducedfrac{278922}{74088}
\]
\end{document}

```
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.