Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both algebraic.

My question is, can we determine, by either adding or multiplying digit-wise $(\mbox{mod }10)$, 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 $(\mbox{mod } n)$, where $n$ can be any digit from 2-10, which resulting numbers are transcendental?