Suppose that n has d divisors and that d is even. The product of the divisors of n is n^d/2, since the divisors come in pairs whose product is n. 

For example, if n=12, then the product of the divisors is (1* 12)(2* 6)(3* 4) = 12^3.

If d is odd, there are (d-1)/2 pairs that give a product of n, as well as the divisor sqrt(n). So the product of the divisors is n^[(d-1)/2]n^1/2 = n^d/2.

For example, if n=16, then the product of the divisors is (1* 16)(2* 8)(4) = 16^(5/2).

To summarize, the product of the positive integers divisors of a positive integer n is n^d/2, where d is the number of divisors of n.

Writing 729 as 3^6, we get n^d/2 = 3^6. This implies n^d = 3^12. 

The possible values of (n,d) are (3,12), (9,6), (27,4), (81,3) and (729,2). 

We find that only in the case (n,d) = (27,4) is d the number of divisors of n, so n=27.
Il prodotto dei divisori interi positivi di un intero positivo n è 729. Trova n.