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We're in an integral domain with unity..... suppose the highest common factor of $x_1$ and $x_2$ = 1 and the highest common factor of $y_1$ and $y_2$ is 1.

If $x_1y_1 = x_2y_2$, can you prove that $x_1$ divides $y_2$?

I am working on a larger problem and somehow I am not inclined to instantly accept this seemingly trivial fact. Any tips?

By the way, since there is no order on this domain, we define highest common factor to mean that any other factor will divide it.

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You can't prove it. For instance, it isn't true for the integral domain Z[x1,x2,y1,y2]/(x1y1 - x2y2). –  zeb Apr 12 '12 at 2:01
makes sense. thanks. –  user18911 Apr 12 '12 at 7:10

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