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#### skatenerd

##### Active member

- Oct 3, 2012

- 114

$$[m]\oplus[n]=[m+n]$$ and $$[m]\odot[n]=[mn].$$I need to verify, among other things, that the multiplication cancellation law fails (if \([m]\odot[j]=[m]\odot[k]\) and \([m]\neq{0}\), then \([j]=[k]\) ).

With a bit of inspection, I began to believe that it in fact does not fail, and I thought I constructed a valid proof showing why, until my professor told me otherwise.

Suppose that \([m]\odot[j]=[m]\odot[k]\). This can be written as \([mk]=[mj]\). This holds if \([mj]=[mk+6p]\) for any integer \(p\).

So we have that

$$mj=mk+6p$$ $$6p=mj-mk$$ $$6p=m(j-k)$$ $$6\frac{p}{m}=j-k$$ \(j=k+6\frac{p}{m}\) where \(\frac{p}{m}\) is an integer. This is the same thing as writing \([j]=[k]\), because \([j]=[k+6\frac{p}{m}]\).

I am pretty sure that this proves the multiplication cancellation law holds but apparently that isn't the case, making me feel like I really don't understand this topic well...

If anybody could help show me where my logic goes south, that would be great. Thanks guys