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Travis Slaysman Thursday March 19, 2013 MAT 280.101 Theorem 1.18: If x is an integer, then x cannot be both even and odd. Proof: We will prove this by contradiction. There exists an x such that x is both even and odd. Assume x is an even integer such that it can be written as x = 2(m) where m is an integer. Assume x is an odd integer such that it can be written as x = 2(n) + 1 where n is an integer. Thus 2(m) = 2(n) +1. Dividing by 2 we see that m = n + (1/2). Theorem 1.17 states that for all rational numbers x and y, if x is an integer and y is not an integer, then n x + y cannot be an integer. This contradicts our original assumption thus proving that if x is an integer, then x cannot be both even and odd. Ж