# Divisibility Principles?

**Number Theory**Level 5

\[\large{ S = \dfrac{(7^p - 2^p)(7^q - 2^q)}{pq} }\]

Find all ordered pairs of primes \((p,q)\) where \(p \leq q\), such that \(S\) is an integer.

If the set of all such primes are \( \{ (p_1, q_1) \ , \ (p_2, q_2) \ ,\ \ldots \ , \ (p_n, q_n) \} \), then find the value of:

\[\large{\sum_{i=1}^n (p_i + q_i) = \ ? }\]