order of elements

Hi, please help. urgent...
thank you so much in advance.


question: what is the order of 3^20(mod 31). Do I use the thm : suppose [a] has order e in z/mz then [a]^d has order e l (d,e) ?

I tried, but it doesn't seem to work corrrectly for me...:(


And also this problem,

find all units in Z/200Z. Do i have to test each number such as 1,3, 7, 9,..to 199? or is there a formula I can use.

Quote:

---

Originally Posted by jenjen

question: what is the order of 3^20(mod 31). Do I use .

---

Not sure what you are asking for.
I think you are asking for the index (or order) of 3 modulo 31?
Or are you asking for the order (or index) of 3^20=3486784401 modulo 31?

Quote:

---

find all units in Z/200Z

---

Consider the ring Z_{200}.
What are its units?
Answer, those are the numbers such that gcd(n,200)=1.
Meaning the numbers relatively prime to it.

I am not really sure. Oh I am sorry, I think that question follows from the question

1: the order for 3 (mod 31) is 30. and then from there, what is the order of 3^20 (mod 31)?

Let n>1.

Theorem Let the order of a modulo n is k for relatively prime a to n. Then the order of a^m is n/d where d=gcd(n,m).

Note, gcd(21,30)=1.
Hence, it has the same order.