1. Determine the smallest natural number that is divisible by 63 and whose sum of digits is equal to 63.
Solution. The smallest natural number whose sum of digits is 63 is 9999999, but it is not divisible by 7, so it is not divisible by 63. This means the desired number must be an eight-digit number and we need to reduce the number of nines since $7 \cdot 9 + a > 63$, where $a$ is the eighth digit. According to this, for the digits of the number, $6 \cdot 9 + a + b = 63$, from which $a + b = 9$. We are looking for the smallest natural number, so let $a = 1$ and $b = 8$. This gives the number 18999999. However, it is not divisible by 7, so it is not divisible by 63. The solution to the problem is the number 19899999.