# Analogy for multiplying modulo N

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?

• Off the top of my head, I'd do multiplication on the clock with "integer * number mod 12" ("I'm going to watch a full season of 15 two-hour shows back-to-back starting at 11. What time will the hour hand be pointing to when I'm done?") and then show that it respects values mod 12, i.e. you could have multiplied by 2*3 instead of 2*15 to answer your question, so instead you could have said "I'm watching*3 mod 12) copies of (2-mod-12) hour shows, how many hours mod 12 will I end on? A bit of a stretch, to be fair. Commented Mar 12, 2019 at 23:56
• It's really the reduction mod n you want to explain, so maybe don't let the values in the binary operation trip you up? Instead, what about just focusing on reducing the result? E.g. I have a bunch of 5-dollar Starbucks giftcards, and I offer to buy grande cafe lattes (3.65 dolllars each) for my friends. I don't want to break a gift card if I don't have to. So, if nine friends step up for this handout, how much actual currency do I need to produce? Answer: $(9×3.65)\equiv_5 2.85$ (Of course, the units don't really match problem to answer, but I'm focusing on the reduction aspect.) Commented Mar 13, 2019 at 2:02
• Hm. I don't think I want to explain the reduction mod N, but rather "you are doing this is normal life". Most of the time, it comes up when I explain "There are other rings/fields than Q,R,C" (mathematicians) or "The word "multiplication" can have many meanings in mathematics" (laywomen/laymen). If it was a real-life-example, I wouldn't mind that units change. Treating multiplication as repeated addition somehow feels cheating. Commented Mar 13, 2019 at 6:54
• Not really real life (discounting standardized tests one might have to take), but maybe something related to finding the units digit of $3^{227}.$ Commented Mar 13, 2019 at 10:58
• Maybe something along the lines of "given that today is Wednesday, what day of the week will it be 873 days from now". Still not real life, but what if town council meetings have to be held on Thursdays and you want the number of days until the first Thursday that occurs at least 60 days from now. Probably this would need to be modified (different scenario, larger number, etc.) to make it something one realistically would use modulo arithmetic for --- John has to visit a doctor regarding periodic treatments for a certain rare condition every 11 days and must take a Taxi on Tuesdays . . . Commented Mar 13, 2019 at 12:09

How about this: I made $$\12$$ per hour for $$7$$ hours. I then bought as many books as I could for $$\$$15 each. How much money did I have left? This would illustrate $$7\cdot 12\equiv ? \pmod {15}$$.