Two numbers $\displaystyle x$ and $\displaystyle y$ are chosen from the set $\displaystyle \{1,2,3.....,5n\}$. Find the probability that $\displaystyle x^4+y^4$ is divisible by $\displaystyle 5$.
Hello,
Note that $\displaystyle (5k+r)^4$ with $\displaystyle r \in \{1,2,3,4\}$ has a remainder 1 in the division by 5.
Therefore, none of these numbers $\displaystyle 5k+r$ would be such that $\displaystyle x^4+y^4$ is divisible by 5.
Remain the multiples of 5. There are n of them in the set. How many ways are there to choose 2 objects randomly among n ? (with replacement)