i) Find (just by trial and error) all solutions of
2x2 is congruent to 1 mod 17
and
2x2 is congruent to 1 mod 23
ii) Now find all solutions of
2x2 is congruent to 1 mod 391
[Note: 391 = 17*23]
Note that if :
$\displaystyle x \equiv y \pmod{m}$
$\displaystyle x \equiv y \pmod{n}$
Then :
$\displaystyle x \equiv y \pmod{mn}$
And your solution follows.
This doesn't always work, I forgot how to do this, I need to check. I think it works anyway if both moduli are prime.