Let \(f_i\) be a \(q_i\)-additive function for \(i=1,2, \dots , l\), and let \(m_1, \dots , m_l\), \(b_1, \dots, b_l\), \(\beta \), \(\gamma \) be positive integers. The authors study the distribution of the number \(\omega (n)\) of prime divisors on the set defined by the congruences \(n \equiv \beta \pmod{\gamma } \) and \(f_i(n)\equiv b_i \pmod{m_i} \). They show that with the proper centering and norming, it has a normal limiting distribution, and they give an estimate of type \( \log \log\log N / \sqrt { \log\log N}\) for the error term when the distribution on \(n\leq N\) is considered. This generalizes the work of \textit{C. Mauduit} and \textit{A. Sárközy} [J. Number Theory 61, 25--38 (1996; Zbl 0868.11004), Acta Arith. 81, 145--173 (1997; Zbl 0887.11008)], who investigated the particular case when the functions \(f_i\) are equal to the sum of digits in base \(q_i\). An important tool is a result of \textit{D.-H. Kim} [J. Number Theory 74, 307--336 (1999; Zbl 0920.11067)] on the distribution of systems of \(q\)-additive functions in residue classes.