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Then by definition, n must have a positive divisor besides 1 and n, say m. Since 1 < m < n, and n was the smallest number greater than 1 without a prime factor, m must have a prime factor, say p. Then p is also a prime factor of n, so we have a contradiction. Therefore, every number greater than 1 has a prime factor. 1 introduces an important strategy in proofs. Notice that to prove that every number greater than 1 had a prime factor, we assumed just the opposite. It was as if we admitted defeat from the very beginning!

2: Multiplication table for Terry’s dance steps Stay FlipRt RotRt FlipLft RotLft Spin Stay FlipRt RotRt FlipLft RotLft Spin Stay FlipRt RotRt FlipLft RotLft Spin FlipRt Stay Spin RotLft FlipLft RotRt RotRt FlipLft RotLft Spin Stay FlipRt FlipLft RotRt FlipRt Stay Spin RotLft RotLft Spin Stay FlipRt RotRt FlipLft Spin RotLft FlipLft RotRt FlipRt Stay puts Terry in the same position as a RotLft.

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