Solution:
we have been asked to find
How many positive integers less than 20 have an odd number of distinct factors?
No prime numbers will have an odd number of distinct factors because those numbers are divisible by 1 and itself.
So the options to check are
4,6,8,9,10,12,14,15,16,18
The factors are as below
[tex] \\ for \ 4,\ we \ have \ 1,2,4\\ \\ for \ 6, \ we \ have \ 1,2,3,6\\ \\ for \ 8, \ we \ have \ 1,2,4,8\\ \\ for \ 9, \ we \ have \ 1,3,9\\ \\ for \ 10, \ we \ have \ 1,2,5,10\\ \\ for \ 12, \ we \ have \ 1,2,3,4,6,12\\ \\ for \ 14, \ we \ have \ 1,2,7,14\\ \\ for \ 15, \ we \ have \ 1,3,5,15\\ \\ for \ 16, \ we \ have \ 1,2,4,8,16\\ \\ for \ 18, \ we \ have \ 1,2,3,6,9,18\\ [/tex]
So we have 4,9 and 16. But we will add 1 also because it has one and only one factor.
So the numbers are 1,4,9,16.
