The identity element of a group

Question

We define the process in Z. Then, is a group. In this group,which is the identity element? The correct answer is the element 10. why ?

Answer 1

Let $e$ be the identity element. Then by definition, we have for any $a \in \mathbb Z$ that: $$ a = a * e = a + e - 10 $$ Solving for $e$ using high school algebra, we obtain $e = 10$, as desired.