# Lesson 21

Odd and Even Numbers

• Let’s explore even and odd numbers.

### 21.1: Math Talk: Evens and Odds

Evaluate mentally.

$$64+88$$

$$65+89$$

$$14 \boldcdot 5$$

$$14\boldcdot 4$$

### 21.2: Always Even, Never Odd

Here are some statements about the sums and products of numbers. For each statement:

• decide whether it is always true, true for some numbers but not others, or never true
• use examples to explain your reasoning
1. Sums:
1. The sum of 2 even numbers is even.
2. The sum of an even number and an odd number is odd.
3. The sum of 2 odd numbers is odd.
2. Products:
1. The product of 2 even numbers is even.
2. The product of an even number and an odd number is odd.
3. The product of 2 odd numbers is odd.

### 21.3: Even + Odd = Odd

How do we know that the sum of an even number and an odd number must be odd? Examine this proof and answer the questions throughout.

Let $$a$$ represent an even number, $$b$$ represent an odd number, and $$s$$ represent the sum $$a + b$$.

1. What does it mean for a number to be even? Odd?

Assume that $$s$$ is even, then we will look for a reason the original statement cannot be true. Since $$a$$ and $$s$$ are even, we can write them as 2 times an integer. Let $$a = 2k$$ and $$s = 2m$$ for some integers $$k$$ and $$m$$.

2. Can this always be done? To convince yourself, write 4 different even numbers. What is the value for $$k$$ for each of your numbers when you set them equal to $$2k$$?

Then we know that $$a + b = s$$ and $$2k + b = 2m$$.

Divide both sides by 2 to get that $$k + \frac{b}{2} = m$$.

Rewrite the equation to get $$\frac{b}{2} = m - k$$.

Since $$m$$ and $$k$$ are integers, then $$\frac{b}{2}$$ must be an integer as well.

3. Is the difference of 2 integers always an integer? Select 4 pairs of integers and subtract them to convince yourself that their difference is always an integer.

4. What does the equation $$\frac{b}{2} = m - k$$ tell us about $$\frac{b}{2}$$? What does that mean about $$b$$?

5. Look back at the original description of $$b$$. What is wrong with what we have discovered?

The logic for everything in the proof works, so the only thing that could’ve gone wrong was our assumption that $$s$$ is even. Therefore, $$s$$ must be odd.