# 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.

## Solution

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 $\frac{9}{25}=\fbox{\textbf{(A)$ (Error compiling LaTeX. ! File ended while scanning use of \fbox .)\frac{1}{25}$}}$ (Error compiling LaTeX. ! Extra }, or forgotten $.).