Determine the least non-negative residue $x$ in
$$11^{1767} \equiv x \pmod {301}$$
Since $\gcd(11,301)$ the by Euler's Theorem we have
$$ 11^{\phi(301)} \equiv 1 \pmod {301} $$
Using $\phi(301)=252$ gives us
$$ 11^{252} \equiv 1 \pmod {301} $$
Noting that $1767 = 7(252)+3$, we have
$$ \begin{align} 11^{1767} & \equiv (11^{252})^7 \times 11^3 \pmod {301} \\ \\ & \equiv 1 \times 1331 \pmod {301} \\ \\ & \equiv 127 \pmod {301} \end{align}$$
And so the least non-negative residue $x$ is 127.