Question

show that U(14)=<3>=<5>?

Answer 1

Note that the units modulo 14 are the numbers which are coprime to 14. That is:\[U(14)=\{ a\in\mathbb{Z}_{14} \mid \gcd(a,n)=1\}\]\[=\{1,3,5,9,11,13\}\]Now you need to show that this set is the same as the cyclic subgroup generated by both 3 and 5 (modulo 14). So start with 3, and start taking powers of 3 until you get repeats mod 14:\[3^1=3,3^2=9,3^3=27\equiv 13\pmod{14}\]etc.

Answer 2

where does 5,,11,13 come from?
because they are prime under 14?