Divisible by a Dozen

     Here is a nice little number trick to show your friends. Just tell them to take out their calculators and follow these steps:

  1. Select any whole number they like and enter it into the calculator.

  2. Find the fourth power of that number, i.e. x4.

  3. Find the square of the original number, i.e. x2.

  4. Subtract the two results just obtained, i.e. x4 - x2.

     You now announce very proudly, & authoritatively, that the resulting difference is divisible by 12. The number in one dozen!

     Further you can say that this will always be the result, no matter what the original number may be. This is sure to amaze your friends, wouldn't you agree?

     However, remember that in mathematics, the existence of many examples does not say that something is always true, just that it might be true. What is required in such a case is a true proof, an algebraic one, that is.

     So your task now before you is to (1) demonstrate this trick with 2 numbers of your choice, and (2) find a good and convincing algebraic proof that the positive difference of the fourth power and square of any integer is divisible by 12.

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