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.