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Section18.3Exercises

We see in Subsection 18.2.2 that \(r\) is not multiplicative. But could some things still be true?

1

Look at the cases where zero is involved. State the broadest possible multiplicativity result you can for this case.

2

Look at the second two examples in Subsection 18.2.2. There seems to be a specific sort of relationship in the precise way in which these examples are not multiplicative. What is that relationship? Can you prove it? (Hint: first compare the results, only then the individual inputs.)

3

For a fixed \(p(x)\), let \(Z_{p(x)}(n)\) be the number of solutions of the polynomial congruence \(p(x)\equiv 0\text{ (mod }n)\). Use facts from earlier in the text to show that this function is multiplicative. Connect this to the question of whether \(-1\in Q_n\).

4

Let the function \(g\) be given by \begin{equation*}\begin{cases}0 & n\text{ is even }\\1 & n\equiv 1\text{ mod}(4)\\-1 & n\equiv 3\text{ (mod }4)\end{cases}\end{equation*} Show that the function \(g(n)\) is multiplicative.