rove that the sum of an even integer and an odd integer is odd.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Direct Proof: Proving the Sum of an Even Integer and an Odd Integer is Odd**

To prove that the sum of an even integer and an odd integer is odd, let's define the problem using basic algebraic expressions.

1. **Definition of Even and Odd Integers:**
   - An even integer can be expressed as \(2n\), where \(n\) is an integer.
   - An odd integer can be expressed as \(2m + 1\), where \(m\) is an integer.

2. **Sum of Even and Odd Integers:**
   - Let the even integer be \(2n\).
   - Let the odd integer be \(2m + 1\).

3. **Calculating the Sum:**
   \[
   (2n) + (2m + 1) = 2n + 2m + 1
   \]

4. **Rearranging the Expression:**
   \[
   2n + 2m + 1 = 2(n + m) + 1
   \]

5. **Analysis of the Result:**
   - The expression \(2(n + m)\) is even, as it is a multiple of 2.
   - Adding 1 to an even number results in an odd number.

6. **Conclusion:**
   - Thus, the sum \(2(n + m) + 1\) is odd.
   
By this direct proof, we have demonstrated that the sum of an even integer and an odd integer is indeed always odd.
Transcribed Image Text:**Direct Proof: Proving the Sum of an Even Integer and an Odd Integer is Odd** To prove that the sum of an even integer and an odd integer is odd, let's define the problem using basic algebraic expressions. 1. **Definition of Even and Odd Integers:** - An even integer can be expressed as \(2n\), where \(n\) is an integer. - An odd integer can be expressed as \(2m + 1\), where \(m\) is an integer. 2. **Sum of Even and Odd Integers:** - Let the even integer be \(2n\). - Let the odd integer be \(2m + 1\). 3. **Calculating the Sum:** \[ (2n) + (2m + 1) = 2n + 2m + 1 \] 4. **Rearranging the Expression:** \[ 2n + 2m + 1 = 2(n + m) + 1 \] 5. **Analysis of the Result:** - The expression \(2(n + m)\) is even, as it is a multiple of 2. - Adding 1 to an even number results in an odd number. 6. **Conclusion:** - Thus, the sum \(2(n + m) + 1\) is odd. By this direct proof, we have demonstrated that the sum of an even integer and an odd integer is indeed always odd.
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