Question

Find the least positive integer which has 26 divisors excluding 1 and the positive integer itself.

Answer 1

P₁ᵃ¹P₂ᵃ² P₃ᵃ³....Pⁿ has (α₁+1)(α₂+1)…(αⁿ+1)divisors, and check minimum integer for each factorization

24=2⋅2⋅2⋅3 are the following:

2⋅2⋅2⋅3 minimum for this factorization is 2²⋅3⋅5⋅7=420

6⋅2⋅2 minimum is 2⁵⋅3⋅5=480

4⋅2⋅3 minimum is 2³⋅32⋅5=360

4⋅6 minimum is 2⁵⋅3³=864

12⋅2 minimum is 2¹¹⋅3>1000

8⋅3 minimum is 2⁷⋅3²>1000

24 minimum is 2²³>1000000

the minimum of the numbers in the right is 360, which is our answer.