**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.8**

Let $Z_{48} = \langle x \rangle$. For which integers a does the map $\varphi_a$ defined by $\varphi_a(1) = x^a$ extend to an isomorphism $\mathbb{Z}/(48) \rightarrow Z_{48}$?

Solution: We know that $\langle x \rangle = \langle x^a \rangle$ precisely when $\mathsf{gcd}(a,n) = 1$. That is, $x^a$ is a generator of $Z_{48}$ precisely when $\mathsf{gcd}(a,n) = 1$. Thus $\varphi_a$ is an isomorphism precisely for 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, and 47.