Find the remainders in the following cases:
(a) $11^{567}$ is divided by 61
(b) $11^{567}$ is divided by 43
(a) We start with
$$ \begin{align} 11^{567} & = 11^{81 \times7} = (11^7)^{81} \\ \\ & \equiv 50^{81} \pmod{61} \\ \\ & \equiv (50^3)^{27} \pmod{61} \\ \\ & \equiv 11^{27} \pmod{61} \\ \\ & \equiv (11^9)^3 \pmod{61} \\ \\ & \equiv 11^3 \pmod{61} \\ \\ & \equiv 50 \pmod{61} \end{align}$$
We used the following to simplify the above.
$ 11^7 = 19487171 \equiv 50 \pmod {61} $
$ 50^3 = 125000 \equiv 11 \pmod {61} $
$ 11^9 = 2357947691 \equiv 11 \pmod {61} $
$ 11^3 = 1331 \equiv 50 \pmod {61} $
So the remainder after $11^{567}$ is divided by 61 is 50.
(b) We start with
$$ \begin{align} 11^{567} & = 11^{3 \times 27 \times 7} = (11^3)^{27 \times 7} \\ \\ & \equiv (-2)^{27 \times 7} \pmod{43} \\ \\ & \equiv (-2^7)^{27} \pmod{43} \\ \\ & \equiv 1^27 \pmod{43} \\ \\ & \equiv 1 \pmod{43} \end{align}$$
We used the following to simplify the above.
$ 11^3 = 1331 \equiv -2 \pmod {43} $
$ (-2)^7 = -128 \equiv 1 \pmod {43} $
So the remainder after $11^{567}$ is divided by 43 is 1.