Divine Remainder

Think of a three digit number, chosen completely at random, and write it down on a piece of paper. Now write the same three digit number again next to it, giving a six digit number all together. I will bet you that I can tell you three numbers which your number will be divisible by, each time leaving no remainder!

First, I will predict that you can divide your number by 7, and find no remainder. Does it work?
Now, you can divide what is left by 11, and again there will be no remainder. Does that work?
Finally, you can take what you now have and divide it by 13. Not only will there be no remainder, but the result of this calculation will amazingly be the three-digit number you first thought of!

How does this work?
If we write a three digit number twice, making a six digit number, then what we have actually done is multiplied our original three-digit number by 1001. To see this, first multiply by 1000, which simply means adding three 0s on the end. Then add your original number once - the three digits added to the three zeros will make the whole six- digit number be the same three digits twice.
The number 1001 has the property that it is made up of three distinct numbers multiplied together: 7 x 11 x 13 = 1001. This means that any number we have created using the method above, when divided by 1001, will go back to the original three digit number. Dividing by 7, then 11, then 13, one after the other, is the same as dividing by 1001.

This trick seems to be very impressive - because this will work for whatever three-digit number you choose, and it must surely be impossible for me to guess what numbers will divide with no remainder, when I don't even know what your number is!