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Is there any proof without words for the identity $GCD (a,b) \cdot LCM(a,b)=ab$ ?

Behzad
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    Probably using prime factorization you could pull that off. – LASV Dec 06 '13 at 15:15
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    Does this qualify: $\min(m,n)+\max(m,n)=m+n$ ? – lhf Dec 06 '13 at 15:16
  • @Luis That would be a formal proof, not a proof without words. – Behzad Dec 06 '13 at 15:18
  • The idea is the same exact as @lhf – LASV Dec 06 '13 at 15:19
  • @lhf Well, I think you're pointing to the method of finding LCM and GCD using prime factorization. But if this method is not given, then you're suggestion won't work. – Behzad Dec 06 '13 at 15:20
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    And by the way, the standard proof does not really use prime factorization. – LASV Dec 06 '13 at 15:21
  • See http://math.stackexchange.com/a/144717/589 for a proof using group theory. – lhf Dec 06 '13 at 15:27
  • @Behzad Apparently you mean a proof without words and without formulas. Which leaves pictures I guess; how does one depict GCD and LCM? I think a more interesting challenge would be a proof with words only (no formulas of any kind, maybe one could tolerate mentioning $a$ and $b$); it might be possible. – Marc van Leeuwen Dec 06 '13 at 15:28
  • @MarcvanLeeuwen That would be nice, too! But for your question: One can imagine $GCD(a,b)$ as the side of the greatest square, covering an $a\times b$ rectangle. Of course the word "greatest$ can not be ommitted! – Behzad Dec 06 '13 at 15:30

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$$ GCD(a,b) = h \rightarrow (a = hr_a) \land (b = hr_b)\land((r_a,h)\times(r_b,h)\times(r_a,r_b)=1) \\ (((((\forall x((x \mid a) \lor (x \mid b) \rightarrow (x \mid g)) \rightarrow ((h \mid g) \land (r_a \mid g) \land (r_b \mid g))) \rightarrow (hr_ar_b \mid g)) \land ((a \mid hr_ar_b) \land b \mid hr_ar_b))) \rightarrow (LCM(a,b)=hr_ar_b)) \rightarrow (GCD(a,b)LCM(a,b)=h\times hr_ar_b =hr_ahr_b=ab) $$ excuse the (informal) symbolism. i have not yet learned to depict the 2-dimensional lattice which would give a wordless explanation for numbers of the type $p^nq^m$, and serve as a basis for imagining the more general picture

David Holden
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    Thanks, but I do know this proof. What I'm looking for is a proof without words (and of course, without so many symbols!). – Behzad Dec 06 '13 at 17:20
  • diagram: 1. a grey square 2. a grey square with a red square attached 3. a grey square with a yellow square attached 4. a grey square with a red square and a yellow square attached – David Holden Dec 06 '13 at 19:22