make a conjecture for the following show work
the sum of an even and odd number.
- the product of two odd numbers.
Conjecture:
1) The sum of an even number and an odd number is always odd because even numbers are like the cool and collected ones, while odd numbers are like the quirky and spontaneous ones. When you mix the two, you'll get a weird and uneven result.
2) The product of two odd numbers is always odd because odd numbers are just oddballs themselves! When you multiply two odd numbers, it's like the oddness multiplies, creating an even odder result. It's odd-squared, you could say!
Conjecture 1: The sum of an even number and an odd number always results in an odd number.
Show work:
Let's consider an even number represented by 2n, where n is an integer, and an odd number represented by 2m+1, where m is an integer.
The sum of these two numbers can be expressed as (2n) + (2m+1).
By combining like terms, we can simplify the expression to 2n + 2m + 1.
Factoring out the common factor of 2, we get 2(n + m) + 1.
Since n + m is also an integer, let's represent it as k, where k is an integer.
Now, we have 2k + 1, which is an odd number.
Therefore, the conjecture is that the sum of an even number and an odd number always results in an odd number.
Conjecture 2: The product of two odd numbers always results in an odd number.
Show work:
Let's consider two odd numbers represented by 2m+1 and 2n+1, where m and n are integers.
The product of these two numbers can be expressed as (2m+1)(2n+1).
Using algebraic multiplication, we can expand the expression to 4mn + 2m + 2n + 1.
Rearranging the terms, we have 2(2mn + m + n) + 1.
Since 2mn + m + n is also an integer, let's represent it as k, where k is an integer.
Now, we have 2k + 1, which is an odd number.
Therefore, the conjecture is that the product of two odd numbers always results in an odd number.
To make a conjecture for the sum of an even and odd number, we can consider the general properties of even and odd numbers.
An even number can be expressed as 2n, where n is an integer, and an odd number can be expressed as 2n + 1, where n is an integer.
So, if we add an even number (2n) and an odd number (2n + 1), the result would be:
(2n) + (2n + 1) = 4n + 1.
Therefore, we can conjecture that the sum of an even and odd number will always be an odd number.
To make a conjecture for the product of two odd numbers, we again consider the general properties of odd numbers.
An odd number can be expressed as 2n + 1, where n is an integer.
So, if we multiply two odd numbers (2n + 1) and (2m + 1), the result would be:
(2n + 1) * (2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1.
Therefore, we can conjecture that the product of two odd numbers will always be an odd number.
I'm sure your teacher wants YOUR conjectures.
Try it out with some examples.
2 + 3 =
4 + 5 =
3 * 5 =
7 * 9 =