# Suppose is a positive rational in its lowest form. Prove that is also in its lowest form.

Now,

=

=

Now note that a common factor of p + 2q and q is also a factor of (p + 2q)−2q = p, therefore must be 1.

Also a common factor of p + 2q and p + q is a factor of both (p + 2q) − ( p + q) = q and 2(p + q) ( p + 2q) = p hence must be 1.

Hence it is in its lowest form.

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