Getting the Number of Divisors
NOTE: After this lesson, it is recommended that you also learn how to get the sum and product of divisors. For the discussion, click here.
Suppose you are asked to find the number of divisors of a large number, for instance, 6534. This process may take you days, if not months to finish. However, there is one easier way you can get how many divisors a number has.
First, you must read and understand the following definition:
In plain English, the number of divisors of a number is the product of the exponents (incremented by 1) of the prime factorization of a number.
Here is an example:
By prime factorization you will be able to derive 22 x 31 x 71. As the above definition says, we must add one to each of the exponents and then multiply them. Thus:
(2+1)(1+1)(1+1) = (3)(2)(2) = 12
Therefore, the number 84 has twelve divisors.
Going back to the previous example:
By prime factorization you will be able to derive 21 x 33 x 112. Using the given definition:
(1+1)(3+1)(2+1) = (2)(4)(3) = 24
Hence, the given number has 24 divisors.
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