How to prove the NP-completeness of MOD-PARTITION MOD-PARTITION: Given a set of integers $A={a_1,...,a_n}$, their weights $w = \{w_1, w_2, \dots, w_n\}$ and the number $k$, does there exist a subset $X$ of $A$ such that: $(\sum_{x \in X} w(x) * x) \mod k \; = \; (\sum_{a \in A \setminus X} w(a) * a) \mod k$?

Can some please provide some idea on how to prove the NP-Completeness of this problem? I saw the proof of the set-partition problem (using a reduction from subset-sum) and I suppose, that this proof maybe will be some little modification of it.
Showing that this your problem is in NP is easy.

To show that your problem is NP-hard, reduce from PARTITION. The reduction simply chooses a large enough modulus $k$. Details left to you.