Need some cleaning advice please

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I noticed on my ceilings from previous owners that there are brown patches. I'm not sure what it is. I've checked the roof for leaks. The ceiling is that stucco style stuff that has all the little pieces on it. That's the best way I know to describe it. I'm not sure how to clean the spots. I tried sweeping dust off one time but those pieces started breaking loose. I don't want to leave bare patches on the ceiling. What would be a good way to get rid of the brown spots?