2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157496We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generalise a recent result of Y. Yi and W. Zhang.Number Theory11A07, 11L07text