a number under 100 and above 1 that has only 5 
factors.

Theorem:
If p is a prime number, then pn has
n+1 factors

So if n+1 = 5
        n = 4

So the 4th power of any prime number has 5 factors.  
There are two 4th powers of primes between 1 and 100,
16 and 81.

24 = 16 has 5 factors, 
namely, 1,2,4,8, and 16
34 = 81 has 5 factors,
namely, 1,3,9,27, and 81

So there are two solutions, 16 and 81.

Edwin