Prove that the square of an odd integer is odd.
Let n be an odd integer. This means that n is 1 more than an even integer. By definition, even integers are multiples of 2 so all even integers can be written in the form 2m where m is an integer. Therefore, n = 1 + 2m.n2 = (1+2m)2 = 1 + 4m + 4m2 = 1 + 2(2m + 2m2)Again, by definition, 2(2m + 2m2) is even. Therefore, n2 is 1 more than an even integer meaning that n2 is also odd.Thus, we have proven what was required.
MO
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