A die (with six sides) is rolled repeatedly and summed. Show that the expected number of rolls until the sum is a multiple of $n$ is $n$.
In
http://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf
problem 12 page 15 a proof is given, assuming that the distribution for the sum of dice values for n > 6 is uniform and equals $\frac{2}{7}$ which is a false assumption that should not be used in a rigorous proof.