Magical Number 1001 - Maths Game to Wow

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I love to break the pleasant monotony of science writing by something that can be really good fun.  
In this blog, I shall talk about a number game and end with describing the magic of 1001.

The game requires the help of a basic four-function calculator - so have one handy.  
Let us start

1. Think of a three digit number - any number that you fancy. For example, let us choose 283.
2. Place this number next to itself to generate a six digit number.  This will be 283283 in our example.
3. Add 343 to this number and divide the sum by 7
4. Subtract 137 from the result and divide by 11
5. Add 8 and divide the result by 13.

What do you get?  In our example, you will get back 283 - the number you chose in the first place.  Cool!!

This will work for any three digit number whatsoever that you choose.  You can also choose 1 or 2 digit numbers so long as you add zeros to the left to make them three digits.  
If you choose 7 then make it 007, or if you choose 53 then make it 053 and the six digit number will be 053053.
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There is a shorter version of this game.  In the simpler version, just divide the six digit number, you have created by 7, 11 and 13 and you retrieve the number you chose in the first place.  It works but it is less fun and carries a smaller wow factor.
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The reason this game works is because of the number 1001 that has for its factors the three prime numbers 7, 11 and 13.  
Also the multiplication of any three digit number by 1001 just makes it a six digit number with the initial number repeated.  By constructing the six digit number, you were effectively multiplying it by 1001.  
Like all good puzzles, it is really simple when explained.

The first puzzle here is something I concocted so that it looks more mysterious but the basic idea is the same.
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Hope you enjoyed the little game.  Let me know.
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