Count The Divisors

There is a certain number \(N = n_1\cdot n_2\cdot n_3\cdots n_k\), where \(1 \leq n_i \leq 100\) \((1\le i \le k)\) are all distinct positive integers.

We are told the following:

  • \(N\) has \(d\) (positive) divisors and \(d\) is odd.
  • \(128\cdot N\) has \(\frac{18}{11}d\) divisors.
  • \(175\cdot N\) has \(\frac 8 5 d\) divisors.
  • \(213\cdot N\) has \(\frac 8 3 d\) divisors.

If \(2016\cdot N\) has \(\frac A B d\) divisors, with \(A\) and \(B\) coprime positive integers, submit \(A + B\).

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