Question: I get the Extended Euclidean Algorithm but I don't understand what this question is asking me:

In this question we consider arithmetic modulo 60 ("clock arithmetic" on the clock with numbers {0,1,2,3,...,58,59}). 
Which of the following statements apply?

a. 50 X 50 = 40 modulo 60

b. 17 + 50 = 3 modulo 60

c. 4100 x 10 = 0 modulo 60

d. 1/17 = 53 modulo 60

e. 37 + 50 = modulo 60

f. 1/10 is undefined modulo 60

g. 17 - 35 = 42 modulo 60

h. 17 - 25 = 18 modulo 60

I have to pick the right ones but I'm not sure how to work it out? Like if they gave something like 

17x = 1 mod(43) I could solve it but I'm not sure how you would solve the other question

P.S I have the answers I just dont want to look at them as I'd rather try to understand first as this is revision for my exam. thanks

Answer: 
You need to differentiate between two different questions:

Q1: Does the right road lead to the ruins?

Q2: Would you say "yes" to question Q1?

If the right road does lead to the ruins, the liar will say "No" to Q1, and *therefore* say "Yes" to Q2!
Since the liar always lies, the only way to extract a truth from a liar is to make him lie a lie, as $\neg (\neg T) = T$.  

Let $p$ be the case that the villager tells the truth.  

Let $q$ be the case that the right branch leads to the ruins.  

We shall prove this by taking cases:

1. The right branch leads to the ruins. (i.e $q$)  

1.1 If $p$, then the answer will be yes and you are good to go.  

1.2 If $\neg p$, if you had asked the liar a question whether $q$, he would say "No" as he wants to lie. However since you've asked him whether he'll answer "yes", if he were not lying then he would have answered "no" but as he always lies, he will answer $\neg$no$=$yes. Thus, you are good to go.
2. The left branch leads to the ruins. (i.e $\neg q$)  

1.1 If $p$, then the answer would be "no" and you should thus take the left branch.  

1.2 If $\neg p$ then the answer would be still be "no" as the villager will lie twice.

Thus if the answer is "no", you should always go left, and if it's "yes" then you should go right.