Answer with explanation:
Let m and n are integers
To prove that m+n and m-n are either both even or both odd.
1. Let m and n are both even
We know that sum of even number is even and difference of even number is even.
Suppose m=4 and n=2
m+n=4+2=6 =Even number
m-n=4-2=2=Even number
Hence, we can say m+n and m-n are both even .
2. Let m and n are odd numbers .
We know that sum of odd numbers is even and difference of odd numbers is even.
Suppose m=7 and n=5
m+n=7+5=12=Even number
m-n=7-5=2=Even number
Hence, m+n and m-n are both even .
3. Let m is odd and n is even.
We know that sum of an odd number and an even number is odd and difference of an odd and an even number is an odd number.
Suppose m=7 , n=4
m+n=7+4=11=Odd number
m-n=7-4=3=Odd number
Hence, m+n and m-n are both odd numbers.
4.Let m is even number and n is odd number .
Suppose m=6, n=3
m+n=6+3=9=Odd number
m-n=6-3=3=Odd number
Hence, m+n and m-n are both odd numbers.
Therefore, we can say for all inetegers m and n , m+n and m-n are either both even or both odd.Hence proved.