We desire to display that if we include two weird numbers, the sum is constantly an even number.

You are watching: Is the sum of two odd numbers always even

Before we also write the really proof, we should convince ourselves that the offered statement has actually some fact to it. We deserve to test the statement with a few examples.

I all set the table below to gather the results of some of the numbers that I supplied to test the statement.


It shows up that the statement, the sum of two odd number is even, is true. However, through simply providing infinitely countless examples perform not constitute proof. That is impossible to list all feasible cases.

Instead, we require to show that the statement hold true because that ALL possible cases. The only means to achieve that is to express an odd number in its basic form. Then, we add the 2 odd numbers created in general kind to acquire a amount of an even number to express in a general kind as well.

To compose the proof of this theorem, you should already have a clear understanding of the general creates of both even and odd numbers.

The number n is even if it can be express as


where k is an integer.

On the other hand, the number n is odd if it deserve to be created as


such the k is some integer.



Note: The objective of brainstorming in creating proof is for us to understand what the theorem is trying to convey; and gather enough information to connect the dots, which will be used to bridge the hypothesis and also the conclusion.

Let’s take two arbitrary odd number 2a + 1 and 2b + 1 wherein a and also b room integers.

Since we space after the sum, we want to add 2a + 1 and 2b + 1.

left( 2a + 1 ight) + left( 2b + 1 ight)

which provides us

left( 2a + 1 ight) + left( 2b + 1 ight) = 2a + 2b + 2.

Notice that us can’t combine 2a and also 2b because they are not comparable terms. However, we room successful in combine the constants, thus 1 + 1 = 2.

What can we carry out next? If you think around it, over there is a usual factor of 2 in 2a + 2b + 2. If we factor out the 2, we achieve 2left( a + b + 1 ight).

What’s next? Well, if we look inside the parenthesis, it’s noticeable that what we have is simply an integer. It might not appear as an creature at very first because we check out a bunch that integers being included together.

Recall the Closure residential or commercial property of Addition because that the collection of integers.

Suppose a and also b belong to the collection of integers. The sum of a and b i m sorry is a+b is also an integer.

In fact, friend can expand this closure residential property of enhancement to an ext than two integers. Because that example, the amount of the integers -7, -1, 0, 4, and also 10 is 6 i m sorry is likewise an integer. Thus,


Going back to where we left off, in 2left( a + b + 1 ight), the expression inside the parenthesis is simply an integer due to the fact that the sum of the integers a, b and 1 is just one more integer. Because that simplicity’s sake, let’s call it essence k.

So then,


That means 2left( a + b + 1 ight) have the right to be express as

2left( a + b + 1 ight) =2k

where 2k is the general kind of an even number. The looks choose we have successfully completed what we want to display that the sum of 2 odds is even.

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THEOREM: The sum of two odd number is an even number.

PROOF: suppose 2a+1 and 2b+1 are any kind of two strange numbers wherein a and b are integers. The amount of these two odd number is left( 2a + 1 ight) + left( 2b + 1 ight). This can be streamlined as 2a + 2b + 2 through combining comparable terms. Factor out the greatest common factor (GCF) the old2 indigenous 2a+2b+2 to get 2left( a + b + 1 ight). Because the sum of integers is just another integer, say integer k, climate k=a+b+1. Through substitution, we have 2left( a + b + 1 ight) = 2k where 2k is clearly the general type of an even number. Therefore, the amount of 2 odd numbers is an even number. ◾️