The authors study the problem of prescribed Hodge numbers of compact Kähler manifolds. As main result, they show that, for any \(m\geq 2\) and \(n\geq 1\), any \(n\)-dimensional Hodge diamond, satisfying the Hodge symmetry and the Serre duality, can be realized by the Hodge numbers of a smooth complex projective variety of dimension \(n\), in the sense of modulo \(m\). The main result implies that there are no polynomial relations among the Hodge numbers of \(n\)-dimensional smooth complex projective varieties up to Hodge symmetry and Serre duality, which answers a question by Kollar in 2012. Furthermore, the proof can also show that any smooth complex projective variety is birational to a smooth complex projective with prescribed inner Hodge numbers modulo \(m\).