A simple calculation will reveal that 72 has 12 factors including 1 and 72. How many natural numbers less than 100 have 12 such factors

Answer 1

To find the number of factors of a number, list its prime factors.
e.g.
12 = 2x2x2x3x3 = (2^3)(3^2)
so now we argue as follows:
the 2's can be used in 4 ways:
that is, we can use none, one, two , or all three of them.
similarly the 2's can be used in 3 ways, none, one of them or both of them
that would include taking neither one ---> represents the factor 1
or taking all, representing the factor 72
then the number of factors including 1 and 72 is 4x3 or 12

So let's work this backwards.
try different combinations of 2's and 3's, staying below a product of 100
e.g. suppose we have 2^5 x 3 , that is 96 which is below 100
we can use the 2's in 6 ways, and the 3 in 2 ways
notice that 6x2 = 12 for 12 factors

So my choice is the number 96, which has 12 factors including 1 and 96 itself

Answer 2

72 = 2^3 * 3^2
So, I'd guess that
96 = 2^5 * 3
will too

Answer 3

the first line of my explanation should have said:
72 = 2x2x2x3x3 = (2^3)(3^2)

Looks like I had 12 on my mind