# Did you count your hops?

**Number Theory**Level 5

Hoppy the Rabbit lives on the vertices of a regular 2015-gon with vertices labelled \(1,2,3,4\dots 2015\). Hoppy lives on vertex 1, and he wishes to visit every vertex this year. To do this he buys a \(k\)-hopping machine. The \(k\)-hopping machine allows Hoppy to hop from his current vertex (assume it has number \(v\)) to the vertex with the number \(v+k\). If \(v+k>2015\) it sends him to vertex \(v+k-2015\) instead. The \(k\) hopping machines exist for \(k=1,2,3\dots 2015\). What is the sum of the values of \(k\) for which the \(k\)-hopping machine allows Hoppy to reach all of the vertices, starting at vertex 1 and hopping a finite number of times?

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