Let $X=\frac{n(5n+1)(10n+1)}{6}$
For $n=1, X= \frac{6*11}{6}=11$
For $n=2, X=\frac{2*11*21}{6}=77$
For $n=3, X=\frac{3*16*31}{6}=248$
For $n=4, X=\frac{4*21*41}{6}=574$
Here, we can see $X$ is not divisible by $3$ and $5$ but $X$ is always an Integer.
Hence, Option(C) Is always an integer.