We note that \(630 = 2 \times 3^2 \times 5 \times 7\). The smallest five 2-digit positive divisors of \(630\) are \(10, 14, 15, 18\), and \(21\). The next largest 2-digit positive divisor is \(30\), and \(21 \times 30 = 630\). Each of \(10, 14, 15\), and \(18\) must be paired with a divisor of \(630\) greater than \(30\). We can check: \(\dfrac{630}{18} = 35\), \(\dfrac{630}{15} = 42\), \(\dfrac{630}{14} = 45\), and \(\dfrac{630}{10} = 63\). Thus, the pairs of 2-digit positive integers whose product is \(630\) are: \((21, 30)\), \((18, 35)\), \((15, 42)\), \((14, 45)\), and \((10, 63)\). There are \(5\) such pairs.