21.13. The numbers $2^{n}$ and $5^{n}$ (in decimal notation) are written one after the other. How many digits does the resulting number have?
21.13. Answer: $n+1$. Suppose the number $2^{n}$ has $p$ digits, and the number $5^{n}$ has $q$ digits. Then $10^{p-1}<2^{n}<10^{p}$ and $10^{q-1}<5^{n}<10^{q}$. Multiplying these inequalities, we get $10^{p+q-2}<10^{n}<10^{p+q}$. Therefore, $p+q=n+1$.