Three digit magic

Saw this amazing trick in one of Smullyan's books.

Puzzle

"Think of any three digit number," says the magician. "Now write that number down twice. So if you thought of 123, then you would right 123123." The participant does so. "Alright, now divide your number by 7." The participant does. "I can tell you that the result is still a whole number! But wait, there's more. Divide the result by 11. And still you have a whole number. Now divide that result by 13. Still a whole number. In fact, you are back to your original three digit number." The volunteer is amazed. Should he be?

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Solution

Certainly not. Dividing by 7, 11, and 13 is the same as dividing by their product, which is 1001. Writing a three-digit number twice is the same as multiplying it by 1001 (the first three digits are multiplied by 1000 and the second three stay the same). So you're simply multiplying by a constant and then dividing by the same constant.

References

The Riddle of Scheherazade: And Other Amazing Puzzles by Raymond Smullyan.