Question 42
Let $p_{1} \lt p_{2} \lt p_{3} \lt p_{4}$ and $q_{1} \lt q_{2} \lt q_{3} \lt q_{4}$ be two sets of prime numbers such that $p_{4} - p_{1} = 8$ and $q_{4} - q_{1} = 8$. Suppose $p_{1} \gt 5$ and $q_{1} \gt 5$. Prove that $30$ divides $p_{1} - q_{1}$.