The number of positive divisors of ${\left(2008\right)}^8$  that are less than ${\left(2008\right)}^4$  are

${\left(2008\right)}^8=2^{24}\times {251}^8$ has $25\times 9=225$   positive divisors, including${\left(2008\right)}^4=\sqrt{{\left(2008\right)}^8}$ . There is a one to one correspondence between the positive divisors less than ${\left(2008\right)}^4$  and those larger than ${\left(2008\right)}^4$ . It follows that there are $\frac{1}{2}\left(225-1\right)=112$ positive divisors less than ${\left(2008\right)}^4$ .