Driving Around In Circles

Nine plots of land are arranged in a circle. Each plot of land has a flower planted on it. A farmer, starting from one of the plots decided to clear all the plots, but in a peculiar way:

• To start, the farmer moves one plot clockwise around the circle and clears that plot of land, then moves two plots clockwise and clears it, then moves three plots clockwise and clears it, and so on.
• If the farmer arrives at a plot of land which he has already cleared, then he will plant a flower there.

How many times would the farmer have circled around all 9 plots of land before he would have cleared all the plots of land?

Image credit: Wikipedia Cropoilbrush