9. Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a, b, c, d$ are non-negative integers.
Solution. The numbers $1,3,5,7,9$ can be represented. For example, $5=\frac{2^{4}-1}{2^{1}-1}$ and $9=\frac{2^{6}-1}{2^{3}-1}$. But 11 cannot. Indeed, Note that the 4 parameters can be simplified into three:
$$
\frac{2^{a}-2^{b}}{2^{c}-2^{d}}=2^{b-d} \frac{2^{a-b}-1}{2^{d-c}-1}=2^{m} \frac{2^{r}-1}{2^{s}-1} .
$$

Since 11 is odd, $m=0$. If we had $11=\frac{2^{r}-1}{2^{s}-1}$, then $11\left(2^{s}-1\right)=2^{r}-1$ which is not possible since all three numbers $11,2^{s}-1$ and $2^{r}-1$ have remainder 3 on division by 4 .