Question

9.
Write down the product of all the divisors of N.​

Answer 1

Answer:

Step-by-step explanation:

The product of the elements in any couple is n, so the product of all the divisors of n is nd(n)/2. Now suppose that n is a perfect square. Then there are d(n)−12 couples plus a solitary individual n1/2. The product of the elements in any couple is n, so the product of all the coupled elements is n(d(n)−1)/2.

Answer 2

Answer:

The function d(n) gives the number of positive divisors of n, including n itself. So for example, d(25)=3, because 25 has three divisors: 1, 5, and 25.

So how do I prove that the product of all of the positive divisors of n (including n itself) is nd(n)2.

For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. d(12) is 6, and 1⋅2⋅3⋅4⋅6⋅12=1728=123=1262=12d(n)2