On a conjecture of Krukenberg and a problem of Dalton and Trifonov

We prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the largest modulus is at least 108. We also prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the least common multiple of the moduli is at least 1440. Finally, we prove that if the smallest modulus of a covering system with distinct moduli is 6, then the least common multiple of the moduli is at least $5040$. The constants $108$, $1440$ and $5040$ are best possible. This resolves a conjecture of Krukenberg, a problem of Dalton and Trifonov, and a generalization thereof.