Primitive roots & indices

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   The group $(\mathbb{Z}/11\mathbb{Z})^{\times}$ of nonzero residues modulo the prime $11$ is cyclic. How many primitive roots (generators) does it have?

2. Warm-up

   The group $(\mathbb{Z}/31\mathbb{Z})^{\times}$ of nonzero residues modulo the prime $31$ is cyclic. How many primitive roots (generators) does it have?

3. Warm-up

   The group $(\mathbb{Z}/29\mathbb{Z})^{\times}$ of nonzero residues modulo the prime $29$ is cyclic. How many primitive roots (generators) does it have?

Problems shown under their original terms: Generated (template, solver-computed).