2021 April MIMC 10 Problems/Problem 14

James randomly choose an ordered pair $(x,y)$ which both $x$ and $y$ are elements in the set $\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$ , $x$ and $y$ are not necessarily distinct, and all of the equations: $x+y$ $x^2+y^2$ $x^4+y^4$ are divisible by $5$ . Find the probability that James can do so.

$\textbf{(A)} ~\frac{1}{25} \qquad\textbf{(B)} ~\frac{2}{45} \qquad\textbf{(C)} ~\frac{11}{225} \qquad\textbf{(D)} ~\frac{4}{75} \qquad\textbf{(E)} ~\frac{13}{225}$

Solution

We can begin by converting all the elements in the set to Modular of $5$ . Then, we realize that all possible elements that can satisfy all the expressions to be divisible by $5$ can only happen if $x$ and $y$ are both $0$ (mod $5)$ . Since $x$ and $y$ are not necessarily distinct, we have $3^2=9$ possible $(x,y)$ . There are total of $15\cdot 15=225$ possible $(x,y)$ , therefore, the probability is $\frac{9}{225}=\boxed{\textbf{(A)} \frac{1}{25}}$ .

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2021 April MIMC 10 Problems/Problem 14

Solution