Exercise 2.1.28 (Last digit of $3^{400}$)

Proof. We search the order of $3$ modulo $10$. By Euler’s theorem, since $3\wedge 10=1$,

where $\varphi (10)=4$. So $3^4\equiv 1(\mathit{mod}10)$ (and $3^2\equiv -1\not\equiv 1(\mathit{mod}10)$ (indeed $2^4=81\equiv 1(\mathit{mod}10)$).

Moreover $3^4\equiv 1(\mathit{mod}10)$ implies $3^{400}={(3^4)}^{100}\equiv 1(\mathit{mod}10)$. Therefore the last digit of $3^{400}$ is $1$. □