Tesselations of multidimensional parallelepipeds (Q1587006)

Say that, for a parallelepiped \(A=[0,a_1]\times\dots\times [0,a_n]\), there exists a tesselation of elements of type \(P=[0,p_1]\times\dots\times [0,p_n]\) with natural positive \(a_i\) and \(p_i\), \(1\leq i\leq n\), if \(A=P_1\cup\dots\cup P_s\), where the parallelepipeds \(P_j,\;1\leq j\leq s\), are isometric to \(P\) and the interiors of these parallelepipeds are pairwise disjoint. The author proves that the parallelepiped \(A\) allows a tesselation of elements of type \([0,p]\times [0,1]\times\dots\times [0,1]\) if and only if \(p\) divides one of the numbers \(a_i\). The author also finds a necessary and sufficient condition for the parallelepiped \(A\) to allow a tesselation of elements of type \(p\) for \(n=2\).