Which elements in $\mathbb{Z}_{12}$ have multiplicative inverses?

Question

Which elements in $\mathbb{Z}_{12}$ have multiplicative inverses? What's the multiplicative inverse of $7$ and $11$?

I'm currently practicing for a small test tomorrow and I'd like to know if I did this right?

The following elements in $\mathbb{Z}_{12}$ have multiplicative inverses (because they are relatively prime to $12$):

$$\left\{5;7;11\right\}$$

"What's the multiplicative inverse of $7$?"

We have to find an element, if we multiply it with $7$ we get $1$ as result. It's $\frac{1}{7}$...?

Answer 1

You are correct that $5,7,11$ are invertible mod $12$, but you're missing $1$. It turns out that $5,7$, and $11$ are all their own inverses mod $12$. This can be verified by hand, or using the Chinese remainder theorem: $$ (\mathbb{Z}/12\mathbb{Z})^{\times}\simeq (\mathbb{Z}/4\mathbb{Z})^{\times}\times (\mathbb{Z}/3\mathbb{Z})^{\times}\simeq C_2\times C_2 $$ where $C_2$ is a cyclic group of order $2$.

Answer 2

Since $$-7(5)+12(3)=1$$

$$7(-5) \equiv 1 \mod 12$$

$$7(7) \equiv 1 \mod 12$$