[size=150]Given: If x is an odd number,[br]Prove: then x[sup]3[/sup] is odd.[br][br]To prove the statement "If x is an odd integer, then x[sup]3[/sup] is odd" using a direct proof, we follow these steps:[br][br]1. Assume that x is an ____________________.[br][br]2. By definition, an odd integer can be expressed as _______________, where k is an integer.[br][br]3. Substitute x with 2k + 1 in the expression for _____________________.[br][br]4. Expanding (2k + 1)[sup]3[/sup] using the binomial theorem, we get: ______________________________.[br][br]5. Notice that every term in the expanded expression is _______________, as each term contains a factor of 2k.[br][br]6. Express the expanded expression as 2m + 1, where m is an ____________________.[br][br]7. Therefore, we have shown that x[sup]3[/sup] can be expressed as an odd integer, as it can be written as __________.[br][br]8. Conclude that if x is an odd integer, then x[sup]3[/sup] is odd, as desired.[/size][br][br][size=150]In this direct proof, we assume the premise that x is an odd integer and then use algebraic manipulations to express x[sup]3[/sup] as an odd integer. This confirms that the statement holds true based on the given assumption.[/size]

[size=150]Use the method of direct proof to prove the following statement:[br]Suppose x, y ∈ Z. If x and y are odd, then xy is odd.[/size]