Groups & cyclic groups

What the problems look like

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

1. Warm-up

   In the additive group $\mathbb Z_{20}$ under addition modulo $20$, consider the element $a = 5$. Find the order of $a$ — the smallest positive integer $k$ such that $k \cdot 5 \equiv 0 \pmod{20}$.

2. Warm-up

   In the additive group $\mathbb Z_{63}$ under addition modulo $63$, consider the element $a = 7$. Find the order of $a$ — the smallest positive integer $k$ such that $k \cdot 7 \equiv 0 \pmod{63}$.

3. Warm-up

   In the additive group $\mathbb Z_{6}$ under addition modulo $6$, consider the element $a = 3$. Find the order of $a$ — the smallest positive integer $k$ such that $k \cdot 3 \equiv 0 \pmod{6}$.

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