Let m and n be two integers. Prove that mn and m+ n are both even if and only if m and n

Problem 10

Let  and  be two integers. Prove that  and  are both even if and only if  and  are both even.

                                                                Step by step solution

Step 1 of 4

Given that  and  are integers.

Consider the following for some integer ‘k’,

     - A number of the form  is always an even number.

     - A number of the form  is always an odd number.

To Prove: The numbers  and  are both even if and only if  and  are both even.

CASE-1: Both  and  are even.

 and  for some integers k and l.

Consider the number ,

Since the term is purely a multiple of 2, it must be an even number.

Consider the other number ,

Since the term  is purely a multiple of 2, it must be an even number.