There's no inherent reason you couldn't, but it'd almost certainly be faster, cheaper, and vastly more reliable to do it with binary logic. In actual practice

There's no inherent reason you couldn't, but it'd almost certainly be faster, cheaper, and vastly more reliable to do it with binary logic. In actual practice what we typically do is select primes that are "near" a power of two, called the Mersenne primes (which have the form 2^N - 1) and Pseudo-Mersenne primes (2^N - a). These have special properties that let us work with them more efficiently. Modern algorithms for these kinds of special cases are branchless, divisionless, and constant time, even if they're not quite as fast as operations in the binary fields.