Be careful about the notation for modular arithmetic. Usually, people write a ≡ b mod n, or [a] = [b], where it is understood that [a] is the congruence class of a modulo n (i.e. the equivalence class of a w.r.t the relation bRc iff b ≡ c mod n).
When you write the decimal expansion of a number, say it with a sentence in plain English, e.g: " Let c0100 + c1101 + ... + ck 10k be the decimal expansion of n " is better than " n = c0100 + c1101 + ... + ck 10k " (for all I know, in the second case, the ci could be arbitrary real numbers. This also makes it implicitly clear that you are using existence and uniqueness of the decimal expansion).