Prooftoys in action
1. The repetitious number
In this parlor trick you ask spectator A to write down any three-digit number, then to copy the number to a new piece of paper, followed by the same digits again, making a six-digit number. Without looking, Instruct spectator A to pass the paper to spectator B without revealing the number to you.
Spectator B is to divide the resulting number by 7, and then pass the result to a spectator C, who divides the result by 11. (Tell them not to worry, there will be no remainder!)
C passes it to yet one final spectator. This last spectator is to divide the result by 13 and then read the resulting quotient. The result will be the original three-digit number.
Problem: Prove that this trick always works regardless of the initially-chosen number.
Hint: The key to solving this problem is finding a simple mathematical statement of it.