Exercise 1. This software determines when \(Z_n\) is the only group of order \(n\) in the case that \(n=pq\) where \(p\) and \(q\) are primes and \(p < q\). Run the software for \(n = 3\times5, 3\times7, 3\times11, 3\times13, 3\times17, 3\times31, 5\times7, 5\times11, 5\times13, 5\times17, 5\times31, 7\times11, 7\times13, 7\times17, 7\times19,\) and \(7\times43\). Conjecture a necessary and sufficient condition about \(p\) and \(q\) for \(Z_{pq}\) to be the only group of order \(pq\), where \(p\) and \(q\) are distinct primes.

Please enter \(p\) and \(q\), the result will show below.

\(p\):
\(q\):