Why study Lowest Common Multiple - LCM

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.

• Adding fractions? – danimal May 4 '15 at 9:08
• Maybe: Desired number of participants in some tournament to allow for the most flexible bracket arrangement or group size. – GDumphart May 4 '15 at 9:10

2 Answers

A real life example of finding LCM is adding two fractions together:

$$\frac14+ \frac16 = \frac3{12} + \frac2{12} = \frac{5}{12}$$

How do you know what the common denominator is? Well, it's the LCM of all of the denominators.

• In real life people have different meaning for the word 'real life'. – P Vanchinathan May 4 '15 at 9:21
• @PVanchinathan I think a vast majority of people had to add fractions at some point in their life, and it had a real life meaning to them. – 5xum May 4 '15 at 9:41
• I was writing that in a flippant mood. But OP has the wording "real life example with impact". In a popular talk with engineering students in audience when a reputed mathematician mentioned that his theorem has applications in algebraic geometry, and number theory, no one really was amused, I had to interrupt him saying that with the word application people associate medical imaging, or mobile communication routing etc and such real-life problems. – P Vanchinathan May 4 '15 at 10:34
• @PVanchinathan That, if you ask me, is very narrow minded thinking. – 5xum May 4 '15 at 10:37
• My aim was to point out the disconnect in the way the word application is interpreted by mathematicians and the other scientists. (possibly this is not the right forum for discussing this). – P Vanchinathan May 4 '15 at 10:40

Alice hit the bulls eye 43 times in 57 attempts; and Bob hit it 48 times in 61 attempts. Who has a good record? Ideally to make them comparable both should have made an equal number of attempts. So we scale them up by maintaining hit rate by expressing the success count for the same number of hits; this requires a common multiple. For economy least common multiple makes sense.

• In "real" life, I never think of this sort of problems using the LCM. Either I compare the decimal form of $43/57$ and $48/61$ or I compare $43\times 61$ v.s. $48\times 57$. – Kim Jong Un May 4 '15 at 9:49
• @KimJongUn: There will still be a pedagogical trouble: both the products have both Alice's and Bob's numbers as a factor. So it is not immediately clear when one number is bigger whose rate should be concluded as better. With LCM we'd be able to see which one is the scaled version of whose. In your first method calculating decimals there is no such interpretational trouble, so it is transparent at the cost of carrying out division instead of multiplication. – P Vanchinathan May 4 '15 at 10:52