  Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers
such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W.
Webb established that for a prime $p\ge 5$ the following variation of Lucas'
Theorem modulo prime powers holds
  $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose
mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}.
  $$ In the proof the authors used their earlier result that present a
generalized version of Lucas' Theorem.
  In this paper we present a a simple inductive proof of the above congruence.
Our proof is based on a classical congruence due to Jacobsthal, and we
additionally use only some well known identities for binomial coefficients.
Moreover, we prove that the assertion is also true for $p=2$ and $p=3$ if in
the above congruence one replace $\lfloor(k-1)/3\rfloor$ by $\lfloor
k/2\rfloor$, and by $\lfloor (k-1)/2\rfloor$, respectively.
  As an application, in terms of Lucas' type congruences, we obtain a new
characterization of Wolstenholme primes.
