Can't seem to figure out this binary operation question

Question

On the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, I define the operation ⊗ to be like our standard multiplication, except that the result is just the last digit of the product. For example, 4 ⊗ 7 = 8, since 4 × 7 = 28, and 8 is the last digit of 28. Noticing that 1 is the identity element for ⊗, find the inverse of each element of the set with respect to ⊗ (or if an element doesn’t have an inverse, explain how you know it doesn't)

Answer 1

The operation you are specifying is equivalent to addition modulo 10. The corresponding inverses are just $(10-the \space number)$. The set with the given operation forms the group $\mathbb{Z}_{10}$. Here, the $\mathbb{Z}_{10}$ corresponds to the group of integers with respect to the operation of addition modulo 10. Therefore, every element has an inverse.

Answer 2

As you notice $0$ is the identity. $0 \oplus a = a \oplus 0 = a$.

Notice that if $a, b \ne 0$ then $0 < a + b \le 18 < 20$ so if $a \oplus b =0$ then $a + b = 10$.

So $b = 10 -a$.

So that's the inverse....

$a^{inverse} = 10 - a$. And $a \oplus a^{inverse} = a \oplus (10 - a) = [10] = 0$.