Abstract Algebra?

Question

Find all of the elements k such that U(25) = . Not sure how to go about this one.

Answer 1

I answered this question before, so I'll post my
previous answer for you here.
You are looking for all the generators of U(25).
In the following all arithmetic will be done mod 25 unless
otherwise specified
and I'll use the theory I need but I won't give the proofs.
You can find them in any good number theory book.
First, let's find a generator, or a primitive root mod 25.
Note that 2² = 4 and 2^4 = 1(mod 5)
so 2 is a primitive root of 5.
Since 2^4 = 16(mod 25), 2 is also a primitive root of 25.
To find all the primitive roots we compute the group
U(20) and raise 2 to each of these powers. Each
of these is also a primitive root of 25.
So U(20) = {1,3,7,9,11,13,17,19).
So all the generators of U(25) are
2
2³ = 8
2^7 = 3
2^9 = 12
2^11 = 23
2^13 = 17
2^17 = 22
and
2^19 = 13.