Prove algebraically that the difference between any two different odd numbers is an even number.
Thanks.
an odd number is of the form 2k+1 where k is any integer. So, subtracting 2m+1 from 2n+1 you get
(2n+1)-(2m+1)
= 2n+1-2m-1
= 2n+2m
= 2(n+m)
any multiple of 2 is even
To prove algebraically that the difference between any two different odd numbers is an even number, we can start by representing two different odd numbers with variables. Let's assume the first odd number is represented by "a" and the second odd number is represented by "b".
According to the definition of odd numbers, we know that any odd number can be written in the form 2n + 1, where "n" is an integer.
So, we can represent "a" as 2m + 1 and "b" as 2n + 1, where "m" and "n" are integers.
Now, let's find the difference between these two numbers:
Difference = a - b
= (2m + 1) - (2n + 1)
= 2m + 1 - 2n - 1
Simplifying further, we get:
= 2m - 2n
Now, we can factor out the common factor 2:
= 2(m - n)
Since "m" and "n" are integers, (m - n) will also be an integer.
Therefore, the difference between any two different odd numbers, represented by "a" and "b" in this case, is always even and can be represented as 2 times an integer.
Hence, we have proven algebraically that the difference between any two different odd numbers is an even number.
To prove algebraically that the difference between any two different odd numbers is an even number, let's consider two odd numbers, which we will denote as n and m. Here's how we can go about proving this statement:
1. Write down the expression for the first odd number, n. Since odd numbers are typically denoted as 2k+1, where k is an integer, we can represent n as n = 2a + 1 for some integer a.
2. Similarly, write down the expression for the second odd number, m. Using the same logic as before, we can represent m as m = 2b + 1 for some integer b.
3. Now, calculate the difference between these two odd numbers. Subtract m from n: n - m = (2a + 1) - (2b + 1).
4. Simplify the expression obtained in step 3 to demonstrate that the difference is an even number. Distribute the negative sign to both terms within the parentheses: n - m = 2a + 1 - 2b - 1.
5. Combine like terms: n - m = 2a - 2b.
6. Factor out a 2 from this expression: n - m = 2(a - b).
7. Finally, since (a - b) is an integer (as it is the difference of two integers), the expression 2(a - b) is also an integer. This confirms that the difference between any two different odd numbers, n and m, is an even number.
Therefore, algebraically we have proven that the difference between any two different odd numbers is an even number.