447. Find the greatest common divisor of the numbers 111111 and 111111111.
$\triangle$ Obviously, each of the given numbers is divisible by 111, i.e., the number 111 is their common divisor. But will 111 be their greatest common divisor?

To answer this question, we will divide the given numbers by 111 and determine whether the resulting quotients are coprime numbers. We will have:

$$
111111=111 \cdot 1001, \quad 1111111111=111 \cdot 1001001
$$

Divide 1001001 by 1001 with a remainder:

$$
1001001=1001 \cdot 1000+1
$$

From this, it is clear that the numbers 1001001 and 1001 are coprime: if they had a greatest common divisor $d>1$, then from the last equation it would follow that the remainder, which is 1, is also divisible by $d$, which is impossible (a smaller natural number cannot be divisible by a larger one). Therefore, the greatest common divisor of the given numbers is 111.

Answer: 111.