Homeopathy is a controversial system of medicine where many medicines use highly diluted substances to treat patients.  Some homeopathic dilutions and medicines are so dilute that they exceed Avagardo's number. 

Avagardo's number is a constant which says substances still have molecules to a certain point.  Any substance diluted past avagardo's number is said to contain nothing other than water.  Homeopaths believe water has a memory so medicines past Avagardo's number still retain the substance in the water. 

Many homeopathic potencies go above Avagardo's number.  Homeopathy believes that more dilute a substance, the more powerful it is.  This is counter intuitive to modern medicine. Homeopathic medicines that go above Avagardo's number are 24X or 12C.  Common remedies used by homeopaths that are below Avagardo's number is 1X,3X,6X, 6C and 12X.  These remedies are generally used when someone initially start a homeopathic treatment.  Another common homeopathic remedy for homeopathy is 30C.  This is well above Avagardo's number. 

Homeopathy was invented before Avagardo's number.  Samuel Hannahman did not really know that this number existed.  Theories existed about this number.  1926 was when the exact number was pretty much found.  If Hannahman knew about this number, would his theories have changed?

Does the lower potencies produce better results in trials?  I found no evidence of this.  I did see a lack of research on whether using lower potencies give patients a better result than higher potencies.

Homeopathy lower potencies seem scientifically possible while other higher potencies exceed Avagardo's number.  However, higher potencies seem to show as good as a result as lower potencies.  More research is needed on this topic.