1. How many divisors of the number $30^{2008}$ are not divisors of the number $20^{2007}$?
Solution. Since $30^{2008}=2^{2008} \cdot 3^{2008} \cdot 5^{2008}$ and $20^{2007}=2^{4014} \cdot 5^{2007}$, then all divisors of $30^{2008}$ that are not divisors of $20^{2007}$ are of the form:

1) $2^{k} \cdot 3^{l} \cdot 5^{m}, l=1, \ldots, 2008, k, m=0,1, \ldots, 2008$ and there are a total of $2008 \cdot 2009^{2}$ or of the form:
2) $2^{k} \cdot 5^{m}, m=2008, k=0,1,2, \ldots, 2008$ and there are $1 \cdot 2009=2009$.

Thus, there are a total of $2008 \cdot 2009^{2}+2009$ divisors of the number $30^{2008}$ that are not divisors of the number $20^{2007$.