Proofs of sum of two even numbers is even

Problem

Show that the sum of two even numbers is also an even number.

Solutions

1. Good for 1/5 mark

2 + 4 = 6      6 + 8 = 14,

2+8 = 10      20 + 30 = 50

24 + 10 = 34.

There’s no two even number I can think of that will NOT result to an even number when added.

2. Good for 3/5 mark

All number than ends in 0, 2, 4, 6, and 8 are even numbers. If you add any two of these, the end digit will still belong to this set. So if you add any two even numbers, the sum will still be even.

Comment: Very good reasoning. I just wish you reason using the algebra you just learned. Try representing the even numbers with 2x next time then reason from there. It’s time you practice to think in terms of algebraic symbols and relationship.

4. Good for 3/5 marks

Let m and n be any integer.

m+n = p => p – n = m (Eq. 1)  

 p – m = n (Eq. 2)

Adding equations (1) and (2): 

2p – (m + n) = m + n

2p = 2(m+n)

Comment: It’s great you attempted to use algebra to think and reason. But there’s a problem with the representation. You are asked to show that when two even numbers are added, the sum is also an even number. Your proof shows that when you double the sum of two numbers, it results to an even number.

5. Good for 5 marks

Let m and n be any integer.

So, 2m and 2n represents two even integers because they are multiples of 2.

2m + 2n = 2 (m+n). The sum is still a multiple of 2 hence even.

Comment: Great! 

David Radcliffe in my LinkedIn group wrote a better version of #5. I would demand it from high school students.

Let m and n be any even integers.
There exist integers a and b such that m = 2a and n = 2b.
Adding the equations gives m + n = 2a + 2b.
So m + n = 2(a+b). The sum is a multiple of 2 hence even.