Working on conjectures. The question is Conjecure: The product of any two odd numbers is _____? It shows several examples of odd numbers x odd numbers and the products are all odd. So I think the right answer is "odd numbers" but is there some formula I should be writing down?

Thank you very much. This is my 2nd day in high school and I'm feeling very frustrated by it all, was feeling like an idiot.

I think you've made the proper conjecture. The product of odd numbers will always be odd.

yes

Well, when it comes to math, I try to balance seriousness with a sprinkle of humor. So, here's a lighthearted take on your conjecture:

Conjecture: The product of any two odd numbers is odd.

Now, to answer your question, no, there isn't really a specific formula you need to write down. It's more of a logical observation. Let's think about it for a moment:

When you multiply two odd numbers, you're essentially adding an odd number to itself multiple times. And we all know that adding odd numbers never results in an even number (unless, of course, you're dealing with sufficiently sneaky numbers).

So, the product of any two odd numbers will always be an odd number. It's like the oddity is contagious!

Just remember, in math, odd times odd equals odd. And if you ask me, that's a pretty odd-orable fact! Keep on conjecturing!

To answer this conjecture, you can observe the patterns using specific examples and then generalize the rule. Let's look at a few examples:

1. Odd number x Odd number = Odd number
- 3 x 5 = 15
- 7 x 9 = 63
- 11 x 13 = 143

2. Even number x Odd number = Odd number
- 2 x 3 = 6
- 4 x 7 = 28
- 6 x 9 = 54

From the examples, we can see that the product of two odd numbers is always an odd number. So it seems that your initial conjecture is correct.

Now, regarding the formula, there is no need to write down a complicated formula to prove this conjecture. It can be proven logically by understanding the properties of odd and even numbers.

An odd number can be represented as 2n + 1, where n is an integer. By multiplying two odd numbers, the result can be expressed as:
(2n + 1) x (2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1

In this expression, the term 2(2nm + n + m) is an even number (since it can be expressed as 2k, where k is an integer). Adding 1 at the end ensures that the result is odd.

Therefore, the product of any two odd numbers is always an odd number.

In summary, you can conclude that your conjecture holds true: the product of any two odd numbers is always an odd number.