1. Does the number 1... (1000 ones) have a ten-digit divisor, all digits of which are different?
Answer: No. Solution: If all the digits of a ten-digit number are different, then their sum is 45. This means that such a number is divisible by 9. The number $1 ... 1$ (1000 ones) with a digit sum of 1000 is not even divisible by 3.