Let \(p\) be an odd prime and consider the equation (*) \(x^p + y^p + z^p = 0\) in the first case (i.e., \(p\nmid xyz)\), where \(x, y, z\) are nonzero integers prime to each other. As the criterion of (*) in the first case, it is well-known that the Kummer congruences \([d^{p-2}\{U_t(v)\}/dv^{p-2}]_{v=0}\equiv 0\pmod p\) and \(B_{2k}[d^{p-1- 2k}\{U_t(v)\}/dv^{p-1-2k}]_{v=0}\equiv 0\pmod p\) \((k=1,2,\ldots, (p-3)/2)\) hold, where \(t\in \{-y/x, -x/y, -z/y, -y/z, -x/z, -z/x\}\), \(U_t(v)=1/(1 - te^v)\) and \(B_{2k}\) is the Bernoulli number with the even index notation. By Mirimanoff's results [see p. 139--148 in \textit{P. Ribenboim}, 13 Lectures on Fermat's Last Theorem. New York etc.: Springer-Verlag (1979; Zbl 0456.10006)] one can easily see that if \(t\not\equiv 0, 1 \pmod p\), then the above congruences are equivalent to \([d^{p-2}\{U_t(v)\}/dv^{p-2}]_{v=0}\equiv 0\pmod p\) and \[ [d^k\{U_t(v)\}/dv^k]_{v=0} [d^{p-2-k}\{U_t(v)\}/dv^{p-2- k}]_{v=0}\equiv 0\pmod p \] \((k=1,2,...,(p-3)/2).\) In this paper, the author discusses the \(p\)-divisibility properties of the numerators of \(B_k\) and \([d^k\{U_t(v)\}/dv^k]_{v=0},\) and derives some consequences from the above congruences.