Number of Positive Divisors of an Integer: Proof

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In summary, the number of positive divisors of an integer n is given by the formula d(n) = (v1 + 1)(v2 + 1)⋅⋅⋅(vk + 1), where n = p1v1p2v2⋅⋅⋅pkvk and p1, p2, ..., pk are prime factors of n. To find the number of positive integral divisors of 2r3s, use the formula (r+1)(s+1). Some strategies for finding divisors include writing out the typical divisor and identifying patterns.
  • #1
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Edit: I'm shifting to a more general question:
If the prime factorization of an integer n is given by

n = p1v1p2v2⋅⋅⋅pkvk

then what would be a proof for the number of positive divisors of n being

d(n) = (v1 + 1)(v2 + 1)⋅⋅⋅(vk + 1)
 
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Gear300 said:
I am supposed to find the number of positive integral divisors of 2r3s. The number is (r+1)(s+1). I tried a number of ways, but I'm not getting the intended answer. Any help?

Hi Gear300! :wink:

Just write out the typical divisor …

what does it look like? :smile:
 
  • #3
In each divisor pik appear with 0 <= k <= vi
 

What is the definition of "Number of Positive Divisors of an Integer"?

The number of positive divisors of an integer is the total number of whole numbers that can evenly divide into the given integer without leaving a remainder.

Why is it important to prove the number of positive divisors of an integer?

Proving the number of positive divisors of an integer is important because it helps us understand the properties and relationships of numbers. It can also be used in various mathematical calculations and problem-solving.

What are the steps involved in proving the number of positive divisors of an integer?

The steps involved in proving the number of positive divisors of an integer may vary depending on the specific proof, but generally it involves using mathematical principles, such as prime factorization, to show that the given integer has a certain number of divisors.

Can the number of positive divisors of an integer be negative?

No, the number of positive divisors of an integer cannot be negative. It is always a positive whole number, including 1 and the integer itself.

How is the number of positive divisors of an integer related to its prime factorization?

The number of positive divisors of an integer is related to its prime factorization because the prime factors of the integer will determine the number of divisors it has. For example, an integer with 3 distinct prime factors will have 8 divisors (2^3), while an integer with 4 distinct prime factors will have 16 divisors (2^4).

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