Fundamental Theorem of Arithmetic

Check the following link for the standard exposition of the proof of this theorem:
http://www.bearnol.pwp.blueyonder.co.uk/Math/fundarit.htm

However a simpler proof might go as follows:
1) every a*b is uniquely = ab
2) therefore every a*(1/b) is uniquely = a/b
3) consider b prime
4) induction completes the proof

I can't see anything wrong with this - and you?