How many factors does 32^2 have ? The OA is 25 and it is reached by transforming 36 into a product of its prime exponents - 2 and 3 --> 2^4 * 3^4. However, I can`t understand why my approach is wrong - 1,2,3,4,6,9,12,18,36 (9 factors) are factors of 36; another 36 will have the same number of factors - which means a total of 18?
= (2^2 * 3^2) ^ 2
= 2^4 * 3^4
So total number of factors = (4+1)*(4+1) = 25
This is a formula: first break the number to all possible prime numbers (along with the powers) Then add one to each power and multiple.
N = a^x*b^y*c^z where a b c are distinct prime numbers and x y and z are corresponding powers then
Total number of factors = (x+1)*(y+1)*(z+1)
Try the same for 36 = 2^2 * 3^2
Total factors = (2+1)*(2+1) = 9 which are [1,2,3,4,6,9,12,18,36]
You just cannot add 9+9 by doing so you are considering (1,2,3,4,6,9,12,18,36 ) twice but there are other factors such as 72 which is not covered in your list of 18.
Hope this makes sense and always remeber the formula at your disposal
