6. [4] The positive integer $n$ is such that the numbers $2^{n}$ and $5^{n}$ start with the same digit when written in decimal notation; determine this common leading digit.
Answer: 3. Note $1=1^{2}<2^{2}<3^{2}<10<4^{2}<\cdots<9^{2}<10^{2}=100$. Divide $2^{n}$ and $5^{n}$ by 10 repeatedly until each is reduced to a decimal number less than 10 but at least 1 ; call the resulting numbers $x$ and $y$. Since $\left(5^{n}\right)\left(2^{n}\right)=10^{n}$, either $x y=1$ or $x y=10$. Because $2^{n}$ and $5^{n}$ begin with the same digit, $x$ and $y$ are bounded by the same pair of adjacent integers. It follows that either $x=y=1$ or $3 \leq x, y<4$. Because $n$ is positive, neither $2^{n}$ nor $5^{n}$ is a perfect power of 10 , so the former is impossible.