Find the elements k that make the statement U(25)=<k> true. Can anyone explain?

Answer 1

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.

Answer 2

It would be easier if you stated the problem more fully.