I am in dire need of help! is an abelian group under multiplication mod . If you need more information, please let me know.
1) Find an element of maximal order in .
2) Let and be integers with and . Prove that
3) Let and be positive integers.
(a) Determine the number of elements of order in .
(b) Compute
Thanks for any and all help! I need help with these by tomorrow night, thanks
Remember a simple fact about cyclic groups. Let and then the order of .
Say that (when you can do yourself). Then it is known that .
Any element in has form either: or .
The order of is simply the order of in .
The order of is simply the order of if , otherwise the order is .
You are trying to find how many elements have order . Well, if then only has order .
If then has order and all so that have order .
The only such which makes this true is . Thus for there are two elements with order .
Now we will consider what happens when .
The elements and so that with are precisely that have this order.
It remains to count the # of so that .
Note thus where because we need .
Thus, where .
There are of course such numbers.
Finally we need to double this number because that is the number for each case: .
We get that the number of elements of order is equal to .
As a check note that when we should have element of order .
Also when we should have elements of order (as we got above).
Thus, eventhough this formula was derived for it works for as well.
We conclude that there are elements of order .