CWMO 2016 Problem 4

For an $n$ -tuple of integers, define a transformation to be: $$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$ Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$ , such that for any $n$ -tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$ , after a finite number of transformations, every element in the of the $n$ -tuple is a multiple of $k$ .