Least Common Multiple Introduction
Jacob Bowen
A fun way to introduce least common multiples is the clapping game. Bring a metronome,
or if you have the time and feel the class can handle the excitement, have them suggest songs
with obvious beats. Select the inoffensive song suggestions, and bring them up on a computer
with speakers. With a metronome or song with an obvious beat, have one group of students
who will clap every 6 beats, and another group of students who will clap every 8 beats. Before
starting the song (and the clapping), have students predict on which beat they expect the
two groups will clap together. Some students are stumped by this question, and others
predict 6 × 8 = 48 beats. But low and behold, on the 24th beat, both groups of students
clap at the same time! This is the least common multiple of 6 and 8, and it illustrates the
concept nicely. Record this result.
Using another song, I like to then try this with two numbers with a different greatest
common divisor. After a few songs and a few different pairs, each with different GCDs, a
pattern will emerge on a table the instructor has been writing on the board:
Song
x
Do Wah Diddy
6
Hey Ya
6
Dancing in the Dark 8
y xy
8 48
9 54
12 96
LCM xy/LCM
24
2
18
3
24
4
This leads naturally into a discussion of greatest common divisors. Listing the songs helps
students remember what happened each time. Additionally, a natural discussion emerges on
if xy and the LCM are ever the same, allowing the instructor to introduce the concept of
relatively prime pairs.
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