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My professor really didn't cover orders and instead just assigned problems.
So for this one question it says
Using the answer from the previous exercise (which is 5), find an element of order 11. So that means I need to find 11^x congruent 1 mod 5 right?
First of all, if an element has order 11, then it means that x^11 = 1 (assuming that "1" is the identity in question), and that x^k is not equal to 1 for 1<k<11. You had the order switched.
Second of all, when you are calculating mod 5, all numbers (except 0) will have order at most 4, so I can't see how it makes sense to speak of order 11 when working mod 5.