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

Question

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

Philoid · Accepted Answer

Now,

=

Now note that a common factor of p + 2q and q is also a factor of (p + 2q)−2⋅q = 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.