SOLUTION: which numbers that are less than 100 have the most factors? How many different prime factors do these numbers have?

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Question 1132327: which numbers that are less than 100 have the most factors? How many different prime factors do these numbers have?
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
I'd reformulate this problem in this way 

    What are the numbers that are less than 100 and have maximal number of highlight%28cross%28factors%29%29 divisors? 
    How many different prime highlight%28cross%28factors%29%29 divisors do these numbers have?

to make the formulation more precise (and more professional).


Solution 

The number  96 = 2%5E5%2A3  has  (1+5)*(1+1) = 6*2 = 12 divisors  

    1, 2,  4,  8, 16, 32,

    3, 6, 12, 24, 48, 96.


It has two prime divisors  2  and  3.



The number  60 = 2%5E2%2A3%2A5  has  (1+2)*(1+1)*(1+1) = 3*2*2 = 12 divisors  

    1,   2,   4, 

    3,   6,  12,

    5,  10,  20,

    15, 30,  60.


It has three prime divisors  2, 3 and 5.


I didn't check that these numbers provide the maximum number of divisors, but I think it is so.

Having this  HINT  from me,  you may check/(or disprove) it on your own.

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Addition :

        1)   I was right with the numbers 60 and 96.

        2)   The numbers  84  and  90  also have  12  divisors each.


        See the table of divisors in this Wikipedia article 
        https://en.wikipedia.org/wiki/Table_of_divisors#1_to_100