For a more conventional algebraic proof, notice that
The product is always an even number times an odd number: if is even, then is odd, but if is odd, then . So is a multiple of 2, and so is a multiple of 6. Therefore, is one more than a multiple of 6, proving the theorem.
I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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