modular arithmetic - modulo large numbers

I have this RSA-crypto question on my exam, to make a long story short, we do not need to go in to all details about the question, but rather the specifics with which I get stuck.

I have no idea how to solve this specific task:

http://s22.postimg.org/rh0vgoaz5/Sk_rmavbild_2015_10_22_kl_22_39_02.png

Sorry that the picture is not uploaded here directly but I don't have enough reputation here to do so, so please click the link.

They are using mod 779 btw.

Now What I don't get is how they can simplify 574^2 to -41. Because from what I see neither Fermat's little theorem or Eulers theorem can be applied here, are they using some other formula? Cause if not, this seems next to impossible to calculate without a calculator, and calculators are not allowed on the best.

Thanks in advance.