Let AB = Im, where A is an

Prove that ${\displaystyle n\mid \varphi (a^n-1)}$. Any natural solution that uses $Aut\left(G\right)$

Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is 2.2$2.2$. What is the probability that there will be) at most 2$2$ such accidents in the next 2$2$ months? 

find all values of k sor which they given matrix augmented matrix to a consistent linear system

Proving that the set of units of a ring is a cyclic group of order 4
The set of units of ${\displaystyle \frac{\mathbb{Z}}{10}\mathbb{Z}}$ is ${\displaystyle \{\bar{1},\bar{3},\bar{7},\bar{9}\}}$, how can I show that this group is cyclic?
My guess is that we need to show that the group can be generated by some element in the set, do I need to show that powers of some element can generate all elements in the other congruence classes?
For example $7^2=49\equiv 9\pm \mathrm{mod}10$, i.e. using 7 we can generate an element in the congruence class of 9, but can not generate 29 for example from any power of 7, so is it sufficent to say that an element is a generator if it generates at least one element in all other congruence classes?

Sharon draws a random card, from a regular deck of cards, and rolls a regular 6 sided die. What is the probability that Sharon draws a card that is a Heart and rolls a 3?

help me answer the question 5.9 a