8. (8 marks) Prove that for all integers m and n, m + n and m-n are either both even or both odd

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.