Gutters are typically made from aluminum, no rust, I think you are seeing mold. Rust is typically a brown color, the photos appear to have black, aka mold. Without being able to see the top of the gutter, I would:
1. Power wash the area.
2. As noted have the gutters cleaned and then make sure they drain correctly.
3. See if the end cap on the gutter is properly sealed to the gutter.
IMHO, this looks like the end cap is leaking and the result is mold in the area.
Hope this helps.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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