About a question of Kannappan and Zhang (Q1781913)

The author answers a question posed by \textit{Pl. Kannappan} and \textit{W. Zhang} [Result. Math. 42, No. 3--4, 277--288 (2002; Zbl 1053.11014)] about the divisibility of binomial coefficients by a given prime \(p\). From the old result of \textit{E. Lucas} [Bull. Soc. Math. Fr. 6, 49--54 (1878; JFM 10.0139.04)] \(\binom{up+a}{vp+b}\equiv\binom{u}{v}\binom{a}{b}\pmod p\) the author deduces that \(\binom{k}{i}\not\equiv0\pmod p\) for every \(i=1,2,\dots,k\) if and only if \(k=p^r-1\) for some integer \(r>0\).