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 $\displaystyle 5$.

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- Sep 27th 2008, 05:38 PMpankajClassical definition of probability
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 $\displaystyle 5$.

- Sep 28th 2008, 02:07 AMMoo
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)