Mathematics: The number of true approximations (factors) of a number and the sum of all approximations

One.

we know that each natural number (excluding 0 and 1) has more than 2 factors, the least of which is prime (also called prime), and the factor of Prime is 1 and itself. The natural number of non-prime numbers is also called composite, and they all contain more than 3 (including 3) factors.
1, how to find a number of how many factors?
For a known natural number, ask how many factors it has, the following methods can be used:
First, the known quantity decomposition factorization, this number into several prime power of the multiplication form, and then the exponent of these prime numbers to add one, and then multiply, the product is the result we want.
For example: Ask 360 how many factors.
Because 360 decomposition factorization can be expressed as: 360=2^3x3^2x5,2, 3, 5 of the exponent is 3, 2, 1, so that the number of factors 360 can be calculated as follows:
(3+1) (2+1) (+) = 24.
We know that the 360 factor has a 1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360 of exactly 24, which shows that the above calculation is correct.
　　 2, how to find the minimum natural number with n factors?" The
also has N (n is a certain number) The natural number can have many different numbers, how to find the minimum number of these numbers?
This is the opposite of the previous question, which is the inverse of the previous one.
For example, what is the minimum number of 24 factors?
According to the previous problem to solve the revelation of the process, you can do this, the first 24 decomposition of the form, the 24 is expressed as a number of multiples of the product, and then the number of each minus 1, as prime numbers 2, 3, 5, 7 ... Index, the number of these exponential multiples of the product, try to calculate the minimum number can be. The method is:
because:24=4x6,  24=3x8, 24=4x3x2,
Now try to find out the target number x:
(1), 24=4x6,4-1=3,6-1=5
  x=2^5x3^3 in these three representations respectively =864
    (2), 24=3x8,3-1=2,8-1=7
x=2^7x3^2=1152
(3) 24=4x3x2,4-1=3, 3-1=2, 2-1=1
x=2^3x3 ^2x5=360
Synthesis (1), (2), (3) know 360 is the smallest of 24 factors.

Two.

6=2 3= (2^1) · (3^1),
So the number of approximate 6:1,2,3,6 A total of 4, can also be counted as follows: (+) (+) =4
All the approximate and 1+3+2+6 can also be counted as follows: (2^0+2^1) (3^0+3^1)

Principle: Because 6 is composed of a 2 a 3, 2 can appear 0 times, 1 times, 3 can appear 0 times, 1 times, so all the sum = (2^0+2^1) (3^0+3^1)

Mathematics: The number of true approximations (factors) of a number and the sum of all approximations