I have no idea how to prove or disprove this statement, can someone help?
"If n is a positive integer and a,b are elements of Z, then there exists an integer x such that ax=b (mod n)"
Do you know what "mod n" means? Saying ax= b (mod n) means that ax= b+ nk for some integer k. What you are asked to prove (or disprove) is that, for any integers a and b, there exist x so that ax= b+ nk for some integer k. That can be rephrased as "for any integers, a, b, and n, there exist two integers, x and k, so that ax- nk= b. Now, for example, suppose a and n are both even but b is odd. What happens?