imc

For two numbers to have a difference that is a multiple of $5$, the numbers must be congruent $\mathrm{mod}5$ (their remainders after division by $5$ must be the same). $0,1,2,3,4$ are the possible values of numbers in $\mathrm{mod}5$. Since there are only $5$ possible values in $\mathrm{mod}5$ and we are picking $6$ numbers, by the Pigeonhole Principle, two of the numbers must be congruent $\mathrm{mod}5$. Therefore the probability that some pair of the $6$ integers has a difference that is a multiple of $5$ is $\boxed{1}$.