2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/108812We prove that if $q$ is a power of a prime $p$ and $p^k$ divides $a$, with $k\ge 0$, then \[ 1+(q-1)\sum_{0\le b(q-1)<a} \binom{a}{b(q-1)}\equiv 0\pmod{p^{k+1}}. \] The special case of this congruence where $q=p$ was proved by Carlitz in 1953 by means of rather deep properties of the Bernoulli numbers. A more direct approach produces our generalization and several related results.11 pagesNumber Theory11B65 (Primary); 05A10, 05A19, 11A07 (Secondary)text