Martin Gardner is a very famous problem proposer. He wrote
the Mathematical Games column in Scientific American for twenty-five years. Here
are two problems from his book My Best Mathematical and Logic Puzzles.
1. Write down any three-digit number, and then repeat the digits in the same order to make a six-digit number (e.g., 394,394). Divide this six-digit number by 7 (there won’t be a remainder), divide the result by 11 (again, there won’t be a remainder), and divide this last result by 13 (there still won’t be a remainder!). Show that the number you now have is equal to the three-digit number that you started with!
2. Only two digits in the following long division problem are revealed. Find the other digits.