Determine the last digit of the following numbers:
(a) $1961^{1961}$
(b) $1023^{1022}$
(c) $2019^{2019}$
(a) We have $1961 \equiv 1 \pmod{10}$ and so by Proposition (3.8)
$$ 1961^{1961} \equiv (1)^{1961} \equiv 1 \pmod {10} $$
So the last digit is 1.
(b) Noting that $1022 =2 \times 511$, we have
$$\begin{align} 1023^{1022} & \equiv (3)^{1022} \pmod {10} \\ \\ & \equiv (3)^{2 \times 511} \pmod {10} \\ \\ & \equiv (3^2)^{511} \\ \\ & \equiv (-1)^511 \\ \\ & \equiv -1 \pmod {10} \\ \\ & \equiv 9 \pmod {10} \end{align}$$
So the last digit is 9.
(c) We have
$$\begin{align} 2019^{2019} & \equiv (-1)^2019 \pmod{10} \\ \\ & \equiv 9 \pmod{10} \end{align}$$
So the last digit is 9.