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?

References

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