Let \(a, b\) be positive integers such that \(1 \leq b < a\).

Let \(p\) be an odd prime number. Find the number of subsets of \(\{1, 2, 3, \dots, ap\}\) with \(bp\) elements such that the sum of the elements of each subset is divisible by \(p\).

Let \(n\) be a positive integer. Find the number of subsets of \(\{1, 2, 3, \dots, an\}\) with \(bn\) elements such that the sum of the elements of each subset is divisible by \(n\).

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