A subset here, a subset there ....

Let \(S(n)\) be the number of subsets \(\{a,b\}\) of the set \(\{1,2,3, .... , 1000\}\) such that \(n | ab\).

Find \(S(13) - 2*S(31)\).

Notes:

  • The elements \(a\) and \(b\) must be distinct.

  • By "\(n | ab\)" I mean that \(n\) divides \(ab\).

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