What is the smallest number that has 20 divisors. can you show me all the numbers that are the divisors please

can you show me all the work on how you got it please. thank you .What is the smallest number that has 20 divisors. can you show me all the numbers that are the divisors please
Interesting problem, and not as easy as it might seem I think. I assume you mean 20 distinct divisors. I think that the best way to approach this would be by considering the prime factorization of this number. Say it is





p_1^{r_1}* ... *p_n^{r_n}





where p_1 ... p_n are all prime numbers. The number of divisors of a number of this form is (r_1+1)*(r_2+1)*...*(r_n+1). If we want 20 divisors then, we could have the follow options for the r's





4,3





4,1,1





Corresponding smallest numbers would be:





2^4*3^3 = 16*27 = 432





2^4*3*5 = 16*15 = 240





So 240 would have to be the answer if we wanted exactly 20 divisors, but it might be possible for a smaller number to have more than 20 divisors. I don't want to think about that right now though, so I'll just say 240.What is the smallest number that has 20 divisors. can you show me all the numbers that are the divisors please
I am assuming that you are asking for the smallest whole number (otherwise negative infinity is pretty small).


This would be the number that has divisors as the number 1 through 20.


This number is 20! (20 factorial) = 20 * 19 * 18 * ... * 2 * 1


20! = 2.433*10^18





This is a very big number indeed.





If you want to be a little trickier. The smallest whole number would be 1*-1*2*-2*3*-3*...*10*-10.


This number is 10!虏 = 10! * 10! = 1.32 *10^13


Something like 13,168,189,440,000

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