Worksheet 11.1.1 Powers mod Powers
Main Question: What is the remainder when you divide \(a^p\) by \(p\text{?}\)
1.
Compute \(a^p\) and its remainder when divided by \(p\text{,}\) for various values of \(a\) and \(p\text{.}\) Everyone should do at least 5, and then share with the group. As a group, discuss any patterns you see and form a conjecture.
Find the remainders when you perform the following divisions. Try different values of \(a\text{.}\) You should first guess what the value is based on your conjecture and then verify (or refute) your guess.
2.
\(a^6\) divided by 10?
3.
\(a^9\) divided by 15?
4.
\(a^{13}\) divided by 21?
5.
\(a^{2321}\) divided by \(2419\text{?}\)
Discuss in your groups: how might we think about the main question here in terms of group theory? What would we need to prove (about groups)?