All positive integers n which equal to n=p_1*p_2, where p_1 and p_2 are distinct primes satisfy the condition in the stem. Because the factors of n in this case would be: 1, p_1, p_2, and n itself, so the product of the factors will be 1*(p_1*p_2)*n=n^2.
(Note that if n=p^3 where p is a prime number also satisfies this condition as the factors of n in this case would be 1, p, p^2 and n itself, so the product of the factors will be 1*(p*p^2)*n=p^3*n=n^2, but we are told that n is not a perfect cube, so this case is out, as well as the case n=1.)
For example if n=6=2*3 --> the product of all the unique positive divisors of 6 will be: 1*2*3*6=6^2;
Or if n=10=2*5 --> the product of all the unique positive divisors of 10 will be: 1*2*5*10=10^2;
Now, take n=10 --> n^2=100 --> the product of all the unique positive divisors of 100 is: 1*2*4*5*10*20*25*50*100=(2*50)*(4*25)*(5*20)*10*100=10^2*10^2*10^2*10*10^2=10^9