Who is asking: Student
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?
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
12 = 1x12
12 = 2x6
12 = 3x4
So the number of factors of 12 is even.
On the other hand if you look at 36 then you get
36 = 1x36
36 = 2x18
36 = 3x12
36 = 4x9
36 = 6x6
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.
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