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

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(52786) (Show Source): You can put this solution on YOUR website!
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I'd reformulate this problem in this way

    What are the numbers that are less than 100 and have maximal number of  divisors? 
    How many different prime  divisors do these numbers have?

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

Solution

The number  96 =   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 =   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.

==================

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

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