1.

What is the order of the element $$3$$ in $$U(20)\text{?}$$

2.

What is the order of the element $$5$$ in $$U(23)\text{?}$$

3.

Find three generators of $$\mathbb Z_8\text{.}$$

4.

Find three generators of the $$5^\text{th}$$ roots of unity.

5.

Show how to compute $$15^{40}\pmod{23}$$ efficiently by hand. Check your answer with Sage.