Yolanda and Zachary are sitting at a round table with a number of other people (possibly 0). They play a game in which they pass a ball around the table, always to the person to the left. The ball starts with Yolanda, and after 2017 passes, Zachary has the ball. After \(k\) additional passes, where \(k\) is some positive integer, Yolanda has the ball again.

For how many integers \(k\) between 1 and 2017 inclusive is it possible that Yolanda and Zachary are sitting next to each other?

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