Prove that is odd if and only if both and are odd. We can write this in mathamatical notation as

Let’s first prove that if both and are odd then their product is odd. An odd number is of the form two times and integer plus one. So let and . Then . Which is odd, by definition, it’s two time an integer — in this case the integer is — plus one.

Now we have to prove things in the other direction. If is odd then so are both and . Let’s do a proof by contradiction. We must assume that either or is even and then prove that is even. Assume is even. Then , which is even. The same holds if n is even. We’ve completed our proof by contradiction and so have proven our original statement.

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