I am wondering if the second Quantum Fourier Transform (QFT) in Shor's algorithm is necessary. I am probably missing a point but it seems that an offset elimination function would suffice to determine the period?
QFT eliminates the offset value and it changes the period from r to M/r, where M is a sufficiently large value >> r^2. What we do normally is to take a measurement and get a multiple of M/r. Doing this a couple of times we take the GCD of those measured values and obtain M/r, and then it is easy to obtain r from it.
But is changing the period necessary if there is no offset? Suppose that the period we are looking for is, say 5. Given a superposition with non-zero values only on the multiples of 5, we have a periodic sequence with period 5. Make a measurement and obtain a random multiple of 5. Do this constant number of times. Since all measured values will be multiples of 5, can't I determine r from it by finding their common divisor? So why do we need QFT? Isn't just an offset elimination function sufficient to find the period?