Modular arithmetic is a great help here.

Let's say that A = 10^k - 1·a_k - 1 + 10^k - 2·a_k - 2 + ... + 10⁰·a₀, where a_i are digits of A. You have to calculate that expression modulo B. Note that you can easily calculate in linear time: just start with 10⁰ = 1 and then multiply by ten (and apply modulo afterwards) until you have all k values. Now the expression is slightly simpler: you only have small numbers, so you can just calculate it straightforwardly, applying modulo after each operation.

Another approach: A = a₀ + 10·(a₂ + 10·(... + 10·(a_k - 1)...)). You start calculating that expression from inside: a_k - 1, then you multiply it by ten and add a_k - 2, and so on until a₀.