Find Smallest Number For Number of Factors

I would like to know how you would fine the **smallest ordinary number** with **exactly** a certain number of factors given, for example:

The number of factors is 6

The **smallest number** with 6 factors is 12

12 =

__So the question is really:__

The number of factors is **n**

The smallest number with **n** factors is **X**

How do you find **X**?

Please feel free to use an example because I think doing it all in algebra will be unreasonably difficult

Re: Smallest number with given number of factors

Let N be any natural number and , where are prime numbers and are natural numbers.

Then number of factors of N is given by

You are given number of factors say k.First step would be to factorize k.That is write k as .

Once you have found out such that , your x would be

Main thing here is that you can multiple possibilities for and you have to optimize for that.

Example:Given k=12,find x.

k=3*2*2 . Corresponding x = =60

k=6*2 . Corresponding x = = 96

k=4*3 . Corresponding x = = 72

So minimum x would be 60.