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In the X4X2 program, we proved empirically that X^{4}X^{2} is always a multiple of 12. Here's a slightly more rigorous proof. 1. Factoring out X^{2} : X^{4}X^{2}=X^{2}(X^{2}1) 2. Factor X^{2}1: X^{4}X^{2}=X^{2}(X1)(X+1) 3a. The expression must be divisible by 4: If X is even then X^{2} is divisible by 4 and the expression is divisible by 4. If X is odd then (X1) 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 X1, 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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