Summary: The \(k\)-ary \(n\)-cube \(Q^k_n\) is one of the most commonly used interconnection topologies for parallel and distributed computing systems. Let \(f(n,m)\) be the minimum number of faulty nodes that make every \((n - m)\)-dimensional subcube \(Q^k_{n-m}\) faulty in \(Q^k_n\) under node-failure models. In this paper, we prove that \(f(n, 0)=1\), \(f(n, 1)=k\) for odd \(k \geq 3\), \(f(n,n - 1) = k^{n - 1}\) for odd \(k \geq 3\), and \(k^m \leq f (n,m) \leq {n-1 \choose m-1}k^m - {n-2 \choose m-1}k^{m-1}\) for odd \(k \geq 3\).