Another Divisors Problem

Let \(D_n\) be the product of all positive divisors of a positive integer \(n.\) \(\big(\)For example, \(D_1 = 1\) and \(D_4 = 1 \times 2 \times 4 = 8.\big)\)

What is the smallest positive integer \(n\) for which \(D_n\) can be written as \[D_n = p^a \times q^b \times r^c \times s^d,\] where \(p, q, r, s\) are four distinct prime numbers and \(a,b,c,d\) are four distinct positive integers?


Bonus 1: Can you solve this for prime numbers \(p\) , \(q\) , \(r\) , \(s\) , \(t\) and integers \(a,b,c,d,e?\)
Bonus 2: Can you generalize for \(p_{1}, ...,p_{n},\) where all primes in the prime factorization are raised to different powers?

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