Understanding the Sum of Odd and Even Numbers

When discussing the sum of numbers, it is crucial to understand the nature of odd and even numbers. This article will explore the sum of three odd numbers and clarify whether such a sum can be even or must be odd.

Properties of Even and Odd Numbers

Let's start by recapping the basic properties of even and odd numbers:

2 even numbers sum to an even number: If you add any two even numbers, the result is always even. 2 odd numbers sum to an even number: The sum of two odd numbers is also always even. 1 even and 1 odd number sum to an odd number: The addition of one even number and one odd number results in an odd number.

Double Each Odd Number

Consider variables a, b, and c, where each is an integer. Multiply each by 2:

2a, 2b, and 2c

Regardless of the initial value, they are now all even numbers because they can be evenly divided by 2:

2a/2 a 2b/2 b 2c/2 c

Adding any two of these even numbers results in an even number:

2a 2b 2ab rarr; 2ab/2 ab 2b 2c 2bc rarr; 2bc/2 bc 2a 2c 2ac rarr; 2ac/2 ac

Adding an odd number to any of these even numbers results in an odd number:

2a 1, 2b 3, 2c 5

Despite this change, adding any two of these modified odd numbers still results in an even number:

(2a 1) (2b 3) 2ab 4 rarr; (2ab 4)/2 ab 2 (2b 3) (2c 5) 2bc 8 rarr; (2bc 8)/2 bc 4 (2a 1) (2c 5) 2ac 6 rarr; (2ac 6)/2 ac 3

Proving the Sum of Three Odd Numbers

Clearly, any two odd numbers added together yield an even number, and an even number can be made odd by adding an odd number. Therefore, the sum of three odd numbers must be odd. To further clarify:

Choose any two odd numbers, n1 and n2, from the given three odd numbers. Their sum is even because odd odd even. Add the remaining odd number, n3. The sum n1 n2 n3 is odd because even odd odd.

Algebraically, let's consider three generic odd numbers a, b, and c where a 2m 1, b 2n 1, and c 2p 1. Their sum is:

a b c (2m 1) (2n 1) (2p 1) 2(m n p) 3 2(m n p 1) - 1

Since m n p 1 is an integer, 2(m n p 1) - 1 is odd.

Conclusion

Summarizing the findings, we establish that the sum of three odd numbers is always odd, not even. This is a fundamental property of arithmetic that holds true despite initial confusion or misconceptions.

Frequently Asked Questions

Q: Can the sum of two odd numbers be even?

No, the sum of two odd numbers is always even.

Q: Can an even number be made odd by adding another even number?

No, adding an even number to another even number always results in an even number.

Q: Can an odd number be made even by adding another odd number?

No, adding an odd number to another odd number always results in an even number.

References

Wikipedia - Parity of zero Math is Fun - Even and Odd Numbers