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Positive Integral Divisors

In this trick, we will find the sum of the positive integral divisors of an integer. If you have not studied the previous trick, Positive Integral Divisors 1 , you need to learn that material before continuing.

The sum of the positive integral divisors of a number is simply that. You take all the positive integral divisors of a number and add them up!

 

Example:

Find the sum of the positive integral divisors of 6.

First, find the positive integral divisors of 6. They are 1, 2, 3, and 6. Next, find their sum. 1 + 2 + 3 + 6 = 12. Therefore, the sum of the positive integral divisors of 6 is 12.

Simply adding all the positive integral divisors will work, but as the number of divisors increases and the numbers we are dealing with get larger, the addition becomes too difficult. Luckily, there's a trick!

 

1 Sum of the Positive Integral Divisors

To find the sum of the positive integral divisors of a number, follow these steps:

  • Find the prime factorization of the number in exponent form.
  • Next generate a set of new numbers, based on the primes. There are two rules for generating this new set:
  • If the prime number p is raised to the first power, then the new number is p + 1.
  • If the prime number p is raised to a power n , which is greater than one, then the new number is .
  • Multiply all these numbers together. The final product is the sum of the positive integral divisors.

 

Example:

Find the sum of the positive integral divisors of 15.

  • Find the prime factorization of 15: 3 1 × 5 1
  • Since both primes are raised to the first power, you add 1 to each prime number and then multiply.
  • (3 + 1) × (5 + 1) = 4 × 6 = 24. The sum of the positive integral divisors of 15 is 24.

 

Example:

Find the sum of the positive integral divisors of 12.

  • Find the prime factorization of 12: 2 2 × 3 1
  • Since 2 is raised to the second power, follow the special formula: .
  • Since 3 is raised to the first power, just add 1: 3 + 1 = 4
  • Multiply: 7 × 4 = 28.

The sum of the positive integral divisors of 12 is 28.

This trick takes lots of practice, but it is invaluable when dealing with larger numbers.

 
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