1. The natural numbers from 1 to 1000 are written on a board, each exactly once. Vasya can erase any two numbers and write down one instead: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is a natural power of ten. What is the largest value it can take? (S. Berlov)
Answer. The fourth. Solution. Example. First, we get $10^{4}=\operatorname{LCM}(16,625)$. Then, sequentially taking $\operatorname{GCD}(1, n)=1$ for all remaining $n$ from 2 to 1000, we leave only $10^{4}$ and 1 on the board, and finally, take $\operatorname{LCM}\left(10^{4}, 1\right)=10^{4}$. Evaluation. Note that if $m$ and $n$ do not divide $5^{5}$, then their GCD and LCM do not divide $5^{5}$ either. Since no natural number from 1 to 1000 divides $5^{5}=3125$, we cannot obtain a power of ten higher than the fourth by operations of taking GCD and LCM.