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I'm curious if, given an integer $n$, there is a general method for finding the orders of the reduced residue classes modulo $n$, as well as how many reduced residue classes correspond to each order.

I take $n=663=3\times13\times17$ as an example. There are $\phi(663)=\phi(3)\phi(13)\phi(17)=384$ reduced residue classes $\pmod{663}$.

Further, $\lambda(663)=\text{lcm}(\phi(3),\phi(13),\phi(17))=48$, which means any any reduced residue has order at most $48$.

However, I'm wondering if I can get more information; namely,

  • All the possible orders of reduced residue classes $\pmod{663}$;
  • How many reduced residue classes there are for each order.

For what it's worth, a computer script gives that the possible orders are {1, 2, 3, 4, 6, 8, 12, 16, 24, 48} and

1 reduced residue classes of order 1
7 reduced residue classes of order 2
2 reduced residue classes of order 3
24 reduced residue classes of order 4
14 reduced residue classes of order 6
32 reduced residue classes of order 8
48 reduced residue classes of order 12
64 reduced residue classes of order 16
64 reduced residue classes of order 24
128 reduced residue classes of order 48

I chose $663$ as an example but my primary question is about how to determine these facts in general. Thank you in advance!

buffle
  • 623
  • $\mathbb Z/663\mathbb Z\cong \mathbb Z/3\mathbb Z\times\mathbb Z/13\mathbb Z\times\mathbb Z/17\mathbb Z$ – J. W. Tanner Feb 16 '21 at 01:42
  • @J.W.Tanner Hi, I'm aware of this fact, though I'm not sure how exactly to use it ... would you be able to elaborate? – buffle Feb 16 '21 at 01:44

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