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=5m+n=7+5=12=Even number m-n=7-5=2=Even numberHence, 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=4m+n=7+4=11=Odd numberm-n=7-4=3=Odd numberHence, m+n and m-n are both odd numbers.4.Let m is even number and n is odd number .Suppose m=6, n=3m+n=6+3=9=Odd numberm-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.