6. (Average) For how many primes $p<50$ is $p^{4}+5 p^{3}+4$ divisible by 5 ?
Answer: 13
Solution: Clearly, the expression is not divisible by 5 when $p=5$. For any prime $p$ other than 5 , note that $p^{4} \equiv 1(\bmod 5)$, so $p^{4}+5 p^{3}+4 \equiv 0(\bmod 5)$. Hence, the problem reduces to counting the number of primes less than 50 other than 5 . There are 13 such primes.