Teaching LCM with real-world examples to motivate students

[matqkks]

What is the most motivating way to introduce LCM of two integers on a first elementary number theory course? I am looking for real life examples of LCM which have an impact. I want to be able to explain to students why they need to study this topic.

 

[Nono713]

Tell them they need to study this topic because it's awesome. End of discussion

But on a serious note, LCM might be a bit too basic to have any memorable "real life impact" on its own. It's like asking for real life examples of addition that have an impact... In my opinion a good way to introduce it would be to talk about linear combinations of the form $Ax + By = C$ for A, B, C fixed integers and x, y integer constants, and asking about its solutions in x and y. The LCM pops up in there if A and B are coprime and $C = 1$, at least... and it let's you push towards Bezout's identity and the GCD as well, if that's useful.. I'm sure there's a "make change in money" real life example that can be applied there... no matter how contrived.

 

[phymat]

You go out to a grocery store to buy sausages and buns for a hot dog party you're hosting. Unfortunately, sausages come in a pack of 6, and buns in a pack of 8.

What is the least number of sausages and buns you need to buy in order to make sure you are not left with a surplus of either sausages or buns?

The answer is LCM(6, 8) = 24 for obvious reasons. You buy 4 packs of sausages, and 3 of buns.

There would be a lot of similar examples where you have to pair up objects and the package sizes are different and you don't want any "wastage".

Another example would be a scenario where you and your friend are going to a restaurant. You have lunch there every fourth day, and he has his lunch there every sixth day. How many days before you meet again for lunch at the same restaurant? The answer again is LCM(4, 6) = 12.