Let \(S\) be the set of all positive integers that are \(1\) more than a multiple of \(10\), so \(S = \{1, 11, 21, 31, 41, \dotsc\}\).

We say that an element \(x\) of the set \(S\) is \(S\)-prime if \(x > 1\) and whenever the elements \(a\) and \(b\) of the set \(S\) satisfy \(ab = x\) we have \(a = 1\) or \(b = 1\).

Are there distinct \(S\)-prime numbers \(a\), \(b\), \(c\) and \(d\) such that \(ab = cd\)?