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In the X4X2 program, we proved empirically that  X4-X2 is always a multiple of 12.  Here's a slightly more rigorous proof.  

1. Factoring out  X2 :   X4-X2=X2(X2-1)

2. Factor X2-1:    X4-X2=X2(X-1)(X+1)

3a. The expression must be divisible by 4: If X is even then X2 is divisible by 4 and the expression is divisible by 4.  If X is odd then (X-1) and (X+1) are even and again the expression is divisible by 4.  Therefore the expression is divisible by 4.

3b.  The expression must be divisible by 3:  The expression has 3 consecutive factors X-1, X, X+1.  Given any 3 consecutive integers, one of the them is divisible by 3.   Proof left as an exercise for the reader.

 4. Since the expression is divisible by 3 and by 4 it is divisible by 12.  Q.E.D.

 

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