Total number of Factors of any number.

Hello Aliens. Well, let me ask you a question before starting my blog. Can you find the total number of factors of any number? For example, let us consider the number 4. We know that the factors of number 4 are (1,2,4). But what if I ask you to find the total number of factors of any bigger number.  

Let's say you are supposed to find the total number of divisors of number 2500. Now of course you won't sit and try all the numbers which divide them and then count them. Instead, here we have a simple formula, for that. But before let's have a look at the example below.

we can write 2500, as 

Now factorizing 2500 into a product of prime numbers. We get.

So now we can see that it has been factorized into a product of its prime factors. So here we just have to add "+1" to the powers of the number and then multiply it to give the total number of positive divisors of the number. That is,

                                    

                                        4+1=5

                                and  2+1=3

So the total number of divisors of 2500 is 5  * 3 =15.

They are : 1,2,4,5,10,20,25,50,100,125,250,500,625,1250,2500.

So 2500 has 15 positive divisors.

Similarly, it has 15 negative divisors too. 

They are : -1,-2,-4,-5,-10,-20,-25,-50,-100,-125,-250,-500,-625,-1250,-2500.

So the total number of divisors of 2500 are 30 including both positive and negative.

Let us have a look at another example 48.

We know. we write 48 as,

Here the power of 2 is 4 and 3 is raises to power 1.

 So  we have, 

4+1=5

1+1=2

So total number of positive divisors of 48 is 5 * 2 = 10. That means  48 has 10 positive and 10 negative factors.

They are : 1,2,3,4,6,8,12,16,24,48 and -1,-2,-3,-4,-6,-8,-12,-16,-24,-48.

 Hence the total number of factors of 48 is 20. 

So we can conclude that if we need to find the total number of factors of any number n, then first write n in the form of the product of prime factors. And then add 1 to the powers of these factors then multiply them. Let's say any number n  can be written as

Then a total number of factors of this number would be nothing but equal to (a+1)(b+1)(c+1).

 Hope you have learned something exciting today. Do share if you find it useful. Till then have a good day.