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Definition: a class of conjugated homotheties (conjugated by a translation) is called a factor.

I really don't understand this definition of a factor. I believe it has something to do with the scaling factor of a homothety, but what does it have to do with translations?

There aren't a lot of online resources on this subject, so I hope I'll get a good explanation here. Is there an insightful example that could help me with the intuition?

Thanks in advance.

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