Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).

My question is, can we determine, by either adding or multiplying digit-wise (mod 10) the digits of $\pi$ and $e$, whether the resulting numbers are transcendental?

Since it might be very difficult to determine whether two individual numbers are transcendental, here is a more general question:

If we add or multiply the digits of $\pi$ or $e$ digit-wise (mod $n$), where $n$ is a digit from 2-10, which of the resulting numbers are transcendental?

Please see a nice explanation of the question which came up in the comments:

@FrancoisZiegler: I believe the idea is to create two new operations, call them $+_{10}$ and $\times_{10}$, which come from identifying an element of $\mathbb R$ as a sequence of elements of $\mathbb Z_{10}$, $x \mapsto (\dotsc, 0, 0, \dotsc, a_n, a_{n - 1}, \dotsc, a_0, a_{-1}, \dotsc)$, via the decimal expansion, and adding or multiplying componentwise. The question then is whether $\pi +_{10} e$ or $\pi \times_{10} e$ are (identified with) transcendental numbers.