Multiplication on $\mathbb{Z}_6$

Question

How can I find all solutions to $[2]x=[4]$ in $\mathbb{Z}_6$?

For example, is $[2]\times 8=[16]=[4]$? is this a right way to solve?

Answer 1

0 1 2 3 4 5    n 
0 2 4 0 2 4    2*n in Z6

You have solutions $x = [2]$ and $x = [5]$.

Answer 2

Do you understand what a residue class is? Working modulo 6, $$\ldots, -12,-6,0,6,12,\ldots$$ are in the same residue class. We call this residue class $[0]$ because it's easier handling the smallest numbers around 0, hence $[0]$ to $[5]$. The numbers I just mentioned are in the same residue class because they are congruent to each other modulo $6$. This is the fancy way of saying it. What it really means is that we can get from any integer in the residue class to another by adding a multiple of $6$. For example, $12 = -12 + 4 \cdot 6$.

Try multiplying the different residue classes (eventually you can create a multiplication table). Remember that $[a][b]=[ab]$. So

$[2][0]=[0], [2][1]=[1], [2][2]=[4], [2][3]=[6]=[0], [2][4]=[8]=[2], [2][5]=[10]=[4]$.

So $[2]$ and $[5]$ are solutions.