Euclid's algorithm can be used not only for integers but for any
You could use the same method for this problem as given in the
Approximations, Euclid's Algorithm and Continued Fractions
finding rational approximations to $\pi$. In this method you write
the process down in terms of continued fractions.
Alternatively, see the article by Benjamin Wardhaugh in the Plus
magazine entitled Euclid's Algorithm and Music which shows a method
for solving this problem. The examples given there use different
numbers. If you use this method it might be easier if you write
down your working using the factorizations of the numbers involved,
rather than writing them out in full.