Lucas Type Theorem Modulo Prime Powers

In this note we prove that {equation*} {np^s\choose mp^s+r}\equiv (-1)^{r-1}r^{-1}(m+1){n\choose m+1}p^s \pmod{p^{s+1}} {equation*} where $p$ is any prime, $n$, $m$, $s$ and $r$ are nonnegative integers such that $n\ge m$, $s\ge 1$, $1\le r\le p^s-1$ and $r$ is not divisible by $p$. We derive a proof by induction using a multiple application of Lucas' theorem and two basic binomial coefficient identities. As an application, we prove that a similar congruence for a prime $p\ge 5$ established in 1992 by D. F. Bailey holds for each prime $p$.