Calculator can't help you (not fully! maybe partially)

By Euler's theorem , as $19$ and $92$ are coprime, we have that

$\large 19^{\phi(92)} \equiv 1 \pmod{92}$ , where $\phi(n)$ is Euler's totient function .

Now as $92 = 4 \times 23$ and these two factors are coprime, we have that

$\phi(92) = \phi(4) \times \phi(23) = 2 \times 22 = 44$ . (Note that $\phi(p) = p - 1$ for any prime $p$ .)

So $\large 19^{92} = 19^{2*44 + 4} \equiv 19^{4} \pmod{92} \equiv 49 \pmod{92}$ ,

where a calculator did come into play, (as noted in the title of the question). So $X^{2} = 49 \Longrightarrow X = 7$ .

Now as $7$ and $92$ are also coprime, we can follow the same steps as above to find that

$\large 7^{92} = 7^{2*44 + 4} \equiv 7^{4} \pmod{92} \equiv \boxed{9} \pmod{92}$ .