Why does 6x+3=1 in ℤ8 have solutions x=1 and x=5?

[Math9999]

Homework Statement

Solve 6x+3=1 in ℤ₈.

Homework Equations

None.

The Attempt at a Solution

6x+3=9
6x=6
x=1
-----------
The answers are x=1, or x=5 in the textbook. But how do I get x=5?

 

[fresh_42]

Homework Statement

Solve 6x+3=1 in ℤ₈.

Homework Equations

None.

The Attempt at a Solution

6x+3=9
6x=6
x=1
-----------
The answers x=1, or x=5 in the textbook. But how do I get x=5?

Since $6 \in \mathbb{Z}_8$ is no unit, we cannot divide by $6$. Thus from $6x=6$ we get $6(x-1)=0$ and we need to find all numbers, for which $6y=0$ in $\mathbb{Z}_8$. Now which multiples of $6$ are divisible by eight?

 

[Math9999]

So from the set [0, 1, 2, 3, 4, 5, 6, 7], we have to choose a number to substitute x, right?

 

[fresh_42]

Yes. And solving $6y=0$ (with $y=x-1$) does this, because by
$$
6y=0 \quad \Leftrightarrow \quad 8\,\vert \, 6y
$$
you can compare the prime factors: three twos on the left and only one on the right.

 

[Math9999]

Thank you.