Some number theory questions:

a is invertible mod n if and only if the gcd of a and n is equal to 1. That give you any clues about the first one?

I think you're right about the first two, but I don't know how to do the other one myself.

If you find out and have spare time, let me know! Thanks.

[latex]5^{2011} \bmod{77} \equiv 5^{2011 \bmod{\phi(77)}}[/latex]
[latex]\equiv 5^{2011 \bmod{60}}[/latex] (Since [latex]5^{1980}.5^{31} \equiv 5^{2011} \bmod{77}[/latex] (Read the Wikipedia article on Modular exponentiation))


2011 mod 60 = 31
[latex]5^{2011} \bmod{77} \equiv 5^{31} \bmod{77}[/latex]

To calculate [latex]5^{31} \bmod{77}[/latex] on your calculator:
Since 5^{31} is too big to display on most calculators, split your equation into terms like 5^3 and 5^4 which can be displayed on your calculator like so:
[latex]5^{31} = (5^4)^7.(5^3)[/latex]

Then, calculate each of these terms mod 77:
For example, to calculate [latex]5^3 \bmod{77}[/latex], do [latex]5^{3} \div 77[/latex] on your calculator. Then subtract the integer part so you're left with the fraction bit. (0.6233766233766233766233766233766233766233766233766233766233...)
Then multiply by 77.

Now solve this:
[latex]\equiv (5^4 \bmod{77})^7.(5^3 \bmod{77}) \bmod{77}[/latex]

Tip: for an easier example, look at Wikipedia's example of 7^222 mod 10 on the Euler-Fermat theorem article.