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Blast into Math! The Euclidean algorithm: a computational recipe
This is correct. Applause for the actors! And for the directors (you and me)!
Exercise: Now it’s time for you to be the director and have some fun with your own “actors.” Make some
examples on your own: choose actors for a , b and c , and “audition” them (make sure that they satisfy
the hypotheses of the theorem: a|b and b|c ). Then, follow the “script” of the proof and see how it plays
out…Don’t forget the applause at the end of the show!
Hello b, my name is “a”. ello b, my name is “a”.
H
I I divide you! divide you!
W
Well, I know something you don't know.ell, I know something you don't know.
I
I divide c! divide c!
There is something I ought to tell you.here is something I ought to tell you.
T
I I divide c as well! divide c as well!
The following lemma has a lot of letters (roles), and when we apply it later, we’ll need a lot of actors to
fill those roles. The show must go on!
Proposition 4.1.7 (Actor Lemma (AL)). Let a, c, t, o, r ∈ Z such that a =0, a|t and a|r. Then
a|(ct + or).
Proof: First, a|t means there is some f ∈ Z such that
t = fa.
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