Math 361, Spring 2019, Assignment 3

Read:

1. Section 20.

Carefully define the following terms, then give one example and one non-example of each:

1. Euler totient function.

Carefully state the following theorems (you need not prove them):

1. Criterion for $[a]\in\mathbb{Z}_n$ to be a unit.
2. Theorem relating the group of units of a direct product to the direct product of the groups of units.
3. Theorem regarding $\phi(ab)$ when $\mathrm{gcd}(a,b)=1$.
4. Formula for $\phi(p^n)$.
5. Procedure to compute $\phi(n)$ in general.
6. Euler's theorem.
7. Fermat's theorem.

Solve the following problems:

1. Section 20, problems 1, 5, 7, 10, 27, and 29 (hint for 29: it suffices to show that every non-zero element of $\mathbb{Z}_{383838}$ satisfies the equation $a^{36}=1$. Use the hint in the book, together with the Chinese Remainder Theorem.)

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