Sometimes I want to explain to laymen/new students/laywomen how addition modulo N works. There are some instructive analogies: Addition on the clock (12), Addition on weekdays (7). They illustrate the idea "N=0" well. However, what I struggle with is to give an analogy for multiplication mod N. Something like "Thursday times Tuesday is Monday" - but it should have some interpretation "in real life". What kind of examples can I use?
Maybe the issue is that if the values you're multiplying have units, then the result of multiplying will have the product of those units. Therefore, your result can't really be equivalent to a value in the base set, because the units are different.
Therefore, if applying this to the real world, I suggest considering the repeated-addition interpretation (scalar multiplication) so you can treat the result as equivalent to something in the base set.