2021 April MIMC 10 Problems/Problem 14
James randomly choose an ordered pair which both and are elements in the set , and are not necessarily distinct, and all of the equations: are divisible by . Find the probability that James can do so.
We can begin by converting all the elements in the set to Modular of . Then, we realize that all possible elements that can satisfy all the expressions to be divisible by can only happen if and are both (mod . Since and are not necessarily distinct, we have possible . There are total of possible , therefore, the probability is .