Final answer:
To prove the product of two odd numbers is odd, represent them as 2m+1 and 2n+1, and their product (2m+1)(2n+1) simplifies to an odd number form, 2(2mn+m+n)+1.
Step-by-step explanation:
To use a direct proof to show that the product of two odd numbers is odd, we need to first define what an odd number is. An odd number is any integer of the form 2k + 1, where k is an integer. Let's call our two odd numbers a and b. Thus, a = 2m + 1 and b = 2n + 1, where m and n are integers.
Now, let's multiply a and b:
• a × b = (2m + 1)(2n + 1)
• a × b = 4mn + 2m + 2n + 1
• a × b = 2(2mn + m + n) + 1
Since 2mn + m + n is an integer (let's call it p), a × b can be written as 2p + 1, which is the form of an odd number. Therefore, the product of two odd numbers a and b is also an odd number.