Quote:
Originally Posted by Raman
For this example, the element 3, over on multiplication yields, generating the elements 1, 3, 9 (mod 1000), although the element 7 (mod 1000) is not being generated at all, although, instead it rather generates the element 507 (mod 1000), although, rather.
AGAIN WHY?
What is being it to be the true reason behind this, rather?

3 has order 100 in Z/1000Z, so it can't generate more than 100 of the 1000 elements. It misses 7, 11, 13, 17, 19, 21, ... as well as all multiples of 2 and 5.