Godel's Beta trick of encoding sequences of numbers to a single number uses the Chinese Remainder theorem (which uses the MOD operator) to essentially construct a primitive integer array.
Why would he use this somewhat clumsy approach instead of the far easier and vastly more efficient method using the MOD and DIV operators?
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Expanding on Misha's answer, remember that Godel is trying to produce a first-order formula in a specific language, and this is an extremely limited syntax. An algorithm which is intuitively simple may nonetheless be quite tedious to "encapsulate" in a formula of this type. By contrast, the $\beta$-function approach has no such difficulties (even though the $\beta$ function itself is a bit odd).
Now you might respond that having a clear algorithm should be a satisfying proxy for a specific first-order formula, but that's a very modern perspective and takes a lot for granted: that type of algorithm/formula conflation is only justified by a general theory of algorithms which postdates Godel (Turing machines were introduced in $1936$ but incompleteness is $1931$) ... and moreover includes the fact that addition and multiplication alone are enough to "implement algorithms in a first-order way" (in the appropriate sense).
Efficiency was not a concern, because nobody was ever going to do concrete calculations with the function. Gödel's $\beta$ function $$ \beta(x,y,z) = x \bmod (1 + y + yz) $$ has the advantage of being easier to describe than any efficient scheme. It requires only one even slightly unusual arithmetic operation (mod), and it has a very short definition.
Compare this to the definition he didn't make of $$ \beta(x,y,z) = \left\lfloor\frac{x}{y^z}\right\rfloor \bmod y $$ which involves exponents and integer division. Its advantage is that the numbers $x$ and $y$ will be smaller to encode a particular sequence... but so what?