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In preparation of a test, there were a few things in class that still confused me on how my instructor got the answer:

Using the following definition for $U_n$:

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There were two examples we did in class that I don't understand clearly:

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And this one:

Order of [3] in $U_{20}$ = 4

 [3]^2 = [9]
 [3]^3 = [7]
 [3]^4 = [1]
 [17] = [-3] = 4

I would really like to know what the process is behind these examples to understand them for my test next week. If possible, how would I start with finding the order of [7] in $U_{40}$?

Thanks for reading, really appreciate the help!

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    I am not sure what you are asking here. Those examples are really very detailed already. – 2017-01-30
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    You compute successive powers of 7, reducing modulo 40 if necessary, until you get 1. The number of iterations is the order. – 2017-01-30
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    @TobiasKildetoft I just don't understand where the answer of 4 comes from the 2nd example – 2017-01-30
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    @AlexProvost so when I'm asked to find the orders of 7 in U40, how far would I go and if there are multiple iterations, how far would I have to go? – 2017-01-30
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    As I've said, you continue until you reach 1. This will necessarily happen, as we are in a finite group. – 2017-01-30

1 Answers 1

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Mod 40, we compute: 7*7 = 49 = 9, 7*7*7 = 343 = 23, 7*7*7*7 = 2401 = 1. Therefore the order of 7 is 4.

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    Thanks for the clear explanation! Didn't know it was that simple! – 2017-01-31