Question:
What is the common name used for numbers that have an odd number of factors? What is the least positive integer that has exactly 13 factors?

Hi Melissa,

Lets look at an example, say 12. The factors of 12 are 1, 2, 3, 4, 6 and 12. Notice that they come in pairs since

So the number of factors of 12 is even.
   On the other hand if you look at 36 then you get

and the number of factors is 9, which is odd. The difference here is that 6 is paired with itself and hence only counts once.
   The point is that if k is a factor of n then there is an integer m so that n = kxm. This adds two to the list of factors (m and k) unless k = m. Thus the number of factors is even unless n = kxk, that is n is a square.
   For your second problem go to the Quandaries and Queries section of Math Central, search in the keyword field for factor and look down the list for an answer.

Cheers,
Penny