The following post describes Zn under exponentiation where n is the product of two primes. Each entry in each exponentiation table here is constructed by raising the row index a to the power of the column index b mod n. Note that since this construction is not Abelian the opposite statement does not hold in its exponentiation tables. The post was inspired by a small discovery I came across while taking CMPUT 210 at the University of Alberta.
Zn is closed with respect to exponentiation since for all a, b in Zn, ab = a*a*a*a*... (b factors) and since Zn is closed with respect to multiplication then a*a*a*a*... (b factors) is also in Zn.
However, Zn under exponentiation is not Abelian since ab is not always congruent to ba mod n, therefore most of what was said about multiplication tables in Z sub n and addition tables in Z sub n doesn't necessarily hold.
In fact, Zn under exponentiation as constructed below where n is the product of two primes is not even a group.
So, is it still possible to generate exponentiation tables for Zn without actually computing the greatest common divisor of (ab, n) for all a and b in Zn?