Much of our work up to this point has been based on the ability to compute a remainder. This is all based on the Division Algorithm.

[[Theorem:Division Algorithm]]
{{page>Problem:Division Algorithm#theorem}}

[[Problem:Division Algorithm Proof]]
{{page>Problem:Division Algorithm Proof#problem}}

If you are unable to complete this problem, then please move on to the next. You may find that reading chapter 0 in your abstract algebra text can help you with this one.

One of our main uses of the division algorithm is modular arithmetic.

[[Definition:Modular Arithmetic]]
{{page>Problem:Modular Arithmetic#definition}}

[[Problem:Remainders Equal Iff Difference Is A Multiple]]
{{page>Problem:Remainders Equal Iff Difference Is A Multiple#problem}}

Many encryption algorithms depend entirely on modular arithmetic. The RSA public key encryption algorithm requires that we compute extremely large powers of rather large numbers. For example, we might need to compute 2348971986871578457358918334698187. This can be an expensive operation, unless we find some patterns to help us. The next problem provides a beginning at doing this.

[[Problem:Computing Powers Modn Conjecture]]
{{page>Problem:Computing Powers Modn Conjecture#problem}}

[[Problem:Disjoint Cycle Notation Practice With Automorphisms Of A Square]]
{{page>Problem:Disjoint Cycle Notation Practice With Automorphisms Of A Square#problem}}

[[Problem:Automorphisms On Several Graphs With 4 Vertices]]
{{page>Problem:Automorphisms On Several Graphs With 4 Vertices#problem}}

We made the following conjecture in class
[[Conjecture:How To Win Scoring When Playing With Simple Shift Permutations]]
{{page>Problem:How To Win Scoring When Playing With Simple Shift Permutations#conjecture}}

[[Problem:Prove this conjecture]]
{{page>Problem:Prove this conjecture#problem}}




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