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 \(ab = cd\)?
We’ve been given a new definition (of \(S\)-prime numbers), so it would be a good idea to try to understand that properly before trying to tackle the question. Can you come up with some examples of numbers that are \(S\)-prime, and also some examples of numbers that are not \(S\)-prime?
For example, \(21\) is \(S\)-prime, because if \(ab = 21\) and \(a\) and \(b\) are both \(1\) more than a multiple of \(10\), then \(a = 1\) or \(b = 1\). But \(121\) is not \(S\)-prime, because \(121 = 11 \times 11\) and \(11\) is a member of \(S\) but does not equal \(1\).
