Question

Prove the product of two odd numbers is always odd __________.

A. By mathematical induction
B. Using graphical analysis
C. Through empirical observation
D. By contradiction

Answer 1

The product of two odd numbers is always odd, and this can be proven using proof by contradiction.

Step-by-step explanation:

To prove that the product of two odd numbers is always odd, we can use proof by contradiction.

1. Assume that the product of two odd numbers, x and y, is even.
2. Since an even number divided by 2 gives an integer, we can write x = 2a+1, and y = 2b+1, where a and b are integers.
3. The product of x and y can be written as (2a+1)(2b+1), which simplifies to 4ab + 2a + 2b + 1.
4. If the product is even, then 4ab + 2a + 2b must be even. However, this cannot be true since an odd number added to an even number always gives an odd number.
5. Therefore, our assumption that the product of two odd numbers is even is incorrect. Thus, the product of two odd numbers is always odd.