A *proper factor* of an integer \(N\) is a positive integer, not \(1\) or \(N\), that divides \(N\).

Show that \(3^2 \times 5^3\) has exactly \(10\) proper factors. Determine how many other integers of the form \(3^m \times 5^n\) (where \(m\) and \(n\) are integers) have exactly \(10\) proper factors.

Let \(N\) be the smallest positive integer that has exactly \(426\) proper factors. Determine \(N\), giving your answer in terms of its prime factors.