My math teacher, Mr Paul Williams, was the one who gave me a neat trick to check if an integer is divisible by 3. I always wonder if it always work. After two years, having obtained some extra knowledge about discrete mathematics, I tried to prove the generality of this trick...
Sun Tzu said, "If you know the enemy and know yourself, you need not fear the result of a hundred battles." Before even writing down what we need to prove, let's try to know it first. "To get a feel of it." so says Mr Goh Aik Hui, my discrete math teacher.
Consider the integer 123. Suppose we want to know if 123 is divisible by 3,
how can we do it quickly?
Simple, add up 1+2+3 = 6. Since 6 is divisible by 3, 123 is also divisible by 3.
Consider 55. 5+5 = 10. Since 10 is not divisible by 3, 55 is not divisible by 3
either.
Here are some more examples:
Trying enough number of times, I was convinced that this DOES work. No matter how big the integer is, the trick works perfectly. If you are not convinced, you can check the proof.
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Any follow-ups, comments or objections to this proof?
Feel free contact me dennyisk@comp.nus.edu.sg. Last edited: Monday, 6 May 2002 |
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