Define $S_r(n)$: digit sum of $n$ in base $r$. For example, $38=(1102)_3,S_3(38)=1+1+0+2=4$. Prove: (a) For any $r>2$, there exists prime $p$, for any positive intenger $n$, $S_{r}(n)\equiv n\mod p$. (b) For any $r>1$ and prime $p$, there exists infinitely many $n$, $S_{r}(n)\equiv n\mod p$.