2. For what least natural $n$ do there exist such natural $a$ and $b$ that

$$
\text { GCD }(a, b)=999 \text { and } \text{LCM}(a, b)=n!
$$

(here $n!=1 \cdot 2 \cdot \ldots \cdot n)$ ?
2. For what smallest natural $n$ do there exist such natural $a$ and $b$ that

$$
\text { GCD }(a, b)=999 \text { and } \text{LCM}(a, b)=n!
$$

(Here $n!=1 \cdot 2 \cdot \ldots \cdot n) ?$

ANSWER. $n=37$.

SOLUTION. Since $\text{LCM}(a, b)$ is divisible by $\text{GCD}(a, b)=999=27 \cdot 37$, and the number 37 is prime, $n \geqslant 37$. On the other hand, for $n=37$ the numbers $a=37!, b=999$ are suitable.