Number Theory

First, let's note that if a number has an odd number of divisors, then it must be a perfect square. 

It also must be a multiple of 8.

So we need a perfect square divisble by 8. 

First, we have

$4*4=16$

However, 16 only has 5 factors, with $1,2,4,8,16$

Next, we have $8*8=64$. However, 64 only has 7 factors, with $1, 2, 4, 8, 16, 32, 64$

144 doesn't work, as it has way too much. 

However, we note that $16*16=256$

256 has exactly 9 divisors, of $1, 2, 4, 8, 16, 32, 64, 128,256$

Thus, the smallest possible number is 256. 

So 256 is our answer. 

Thanks! :)