 Solution

From every vertex choose one neighbour of the successor color and follow these choices. The finite directed graph with one chosen outgoing edge per vertex contains a directed cycle. Its vertices are distinct until the first repeated vertex. Each step increases the color by one modulo five, so returning to the starting vertex requires the cycle length to be divisible by five. It therefore has length at least five and is also a cycle in the underlying graph. This proves a cycle of length a positive multiple of five exists.