The Sum and Product of Divisors
NOTE: It is easier to understand the discussion below if you already know how to get the number of divisors of a given number. For the discussion, click here.
Getting the sum and product of divisors can be as stress-free as knowing how many divisors a number has. Just keep in mind the following definitions:
Now let us apply these definitions to the two examples we had in deriving the number of divisors:
Example 1: 84
Recall the prime factorization of 84, which is 22 x 31 x 71. Using the rule on getting the sum of divisors, we get the following values:
The product of 7, 4 and 8 is 224. Thus it can be said that the sum of the divisors of 84 is 224.
This time, let us get the product of the divisors. To simplify the definition of the said process, just divide the number of divisors by 2 and apply it as an exponent to the given number.
Recall that 84 has twelve divisors. We all know that 12 ¸ 2 is 6. Use it as an exponent to 84, and you will get your answer. Hence the product of all the divisors of 84 is 846, or 351,298,031,616.
Example 2: 6534
The prime factorization of 6534 is 21 x 33 x 112. Using the given definition, let us find the sum of divisors:
The product of 3, 40 and 133 is 15960, which is your answer.
The given number has 24 divisors. Thus the product of divisors is 653412, a very big value when expanded (so we'll keep it that way).
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