# Difference between revisions of "2021 April MIMC 10 Problems/Problem 14"

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<math>\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}</math> | <math>\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}</math> | ||

==Solution== | ==Solution== | ||

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

## Revision as of 13:44, 26 April 2021

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 .