10.3. Can two digits be appended to the number 277 so that the resulting number is divisible by any number from 2 to 12?
10.3. For a number to be divisible by 10, it must end in zero. Let's choose the second-to-last digit so that the resulting number is divisible by 9. In this case, the sum of the digits of this number should be equal to 18. Therefore, we get the number 27720. The problem has a solution only if the found number is divisible by each of the numbers 7, 8, and 11. It is easy to check that this is indeed the case. This means that the number 27720 is divisible by any number from 2 to 12 inclusive.