Here is my question:
Consider the subset $S = \{0, 2, 4, 6, 8, 10, 12\}$ in $\mathbb Z_{14}$, with the operations of addition and multiplication in $\mathbb Z_{14}$.
(a) Show that $S$ has a multiplicative identity: that is, find a specific element $x$ of $S$ for which $x\cdot a = a$ for each element $a \in S$.
• Make a multiplication table for $S$. Remember that the multiplication is in $\mathbb Z_{14}$. Your table should contain only the elements ${0, 2, 4, 6, 8, 10, 12}$. Look at the table to find the identity.
(b) Using the multiplicative identity you found in part (a), what is $10^{−1}$? That is, what element multiplied by $10$ gives your multiplicative identity?
For part (a), I found the answer, $x=8$. For part (b), I am stuck, even though it's probably simple.
So, I need to find where $10X = 1$ in $\mathbb Z_{14}$, correct? But anything in $\mathbb Z_{14}$ multiplies by 10 will be even, and therefore will not be $1$. Or am I missing something? Thanks!
You want $10 x = e$ where $e$ is the answer to $(a)$