Even and Odd Number Rules - Third Grade Mathematics

Numbers have distinct personalities, and one of the most fundamental ways we classify whole numbers is into even numbers and odd numbers. In this chapter, you will discover what makes a number even or odd, how pairing and sharing explain this division, and the powerful addition and multiplication rules that predict whether an answer will be even or odd before you even finish calculating.

What Are Even and Odd Numbers?

An even number is any whole number that can be split into two equal groups with zero leftovers, or arranged into pairs where every item has a partner.

An odd number is any whole number that cannot be split into two equal groups evenly; when you try to pair its items, there is always exactly one leftover item without a partner.

Even Number (6):                Odd Number (7):
Pairs:                          Pairs:
[o] [o]   <- Pair 1            [o] [o]   <- Pair 1
[o] [o]   <- Pair 2            [o] [o]   <- Pair 2
[o] [o]   <- Pair 3            [o] [o]   <- Pair 3
All paired up!                 [o]       <- One left out!

The Magic Rule: Look at the Ones Digit

No matter how huge a number is—whether it is 8, 48, 528, or 9,998—you only ever need to inspect one single digit: the digit in the ones place!

Even Numbers end in:  0,  2,  4,  6,  or  8
Odd Numbers end in:   1,  3,  5,  7,  or  9

Let us test some large numbers:
- 3,456 ends in 6, so it is an even number.

- 7,891 ends in 1, so it is an odd number.

- 5,000 ends in 0, so it is an even number.

- 9,235 ends in 5, so it is an odd number.


The thousands, hundreds, and tens digits do not matter at all when deciding if a whole number is even or odd. That is because every group of ten, hundred, or thousand is already made of tens, and ten can always be split evenly in half (5 + 5 = 10)! Only the remaining ones place determines the final balance.

Rules for Adding Even and Odd Numbers

When you add two numbers together, their oddness or evenness follows strict, predictable patterns.

+---------------------+-------------------+---------------------+
| Addition Rule       | Example           | Why It Works        |
+---------------------+-------------------+---------------------+
| Even + Even = Even  | 4 + 6 = 10        | All items paired    |
| Odd + Odd = Even    | 3 + 5 = 8         | Two loners pair up! |
| Even + Odd = Odd    | 4 + 5 = 9         | One loner remains   |
| Odd + Even = Odd    | 7 + 2 = 9         | One loner remains   |
+---------------------+-------------------+---------------------+

Why Does Odd + Odd Equal Even?

Think about the visual model of pairing.
- The first odd number has all pairs plus 1 lone item.

- The second odd number has all pairs plus 1 lone item.

- When you combine the two numbers, the two lone items find each other and form a brand-new complete pair!
Because every item now has a partner, the sum is always even.

Odd (3):   [o][o]  +  [o]   (1 loner)
Odd (5):   [o][o] [o][o]  +  [o]   (1 loner)
Combined:  All existing pairs + [o][o] (new pair formed!) = 8 (Even)

Rules for Subtracting Even and Odd Numbers

Subtraction follows the exact same patterns as addition:
- Even - Even = Even (for example, 10 - 4 = 6)

- Odd - Odd = Even (for example, 9 - 3 = 6)

- Even - Odd = Odd (for example, 8 - 3 = 5)

- Odd - Even = Odd (for example, 7 - 4 = 3)


Rules for Multiplying Even and Odd Numbers

Later in this book, you will study multiplication in detail, but you can already predict the outcome using parity (evenness and oddness):
- Even x Even = Even (for example, 4 x 2 = 8)

- Even x Odd = Even (for example, 4 x 3 = 12)

- Odd x Even = Even (for example, 5 x 2 = 10)

- Odd x Odd = Odd (for example, 3 x 5 = 15)


Notice that as long as at least one factor is even, the product will always be even! Only multiplying two odd numbers produces an odd product.

Chapter Practice Exercises

  1. Classify each number as even or odd: a. 24 b. 57 c. 380 d. 1,495 e. 6,782
  2. Without calculating the exact sum, predict whether the answer is even or odd: a. 34 + 18 b. 45 + 23 c. 68 + 19 d. 101 + 204
  3. Explain using drawings or words why the sum of 5 and 7 must be an even number.
  4. Carlos says: "Every time you add 1 to an even number, you get an odd number." Is Carlos correct? Explain why.
  5. If you have an even number of socks in your laundry basket, can you pair every sock up without any sock left alone? What if you have 17 socks?

Solutions and Step-by-Step Answers

  1. Classification by ones digit: a. 24 ends in 4 -> Even. b. 57 ends in 7 -> Odd. c. 380 ends in 0 -> Even. d. 1,495 ends in 5 -> Odd. e. 6,782 ends in 2 -> Even.
  2. Predictions: a. 34 (Even) + 18 (Even) = Even. b. 45 (Odd) + 23 (Odd) = Even (two odds combine to make an even sum). c. 68 (Even) + 19 (Odd) = Odd. d. 101 (Odd) + 204 (Even) = Odd.
  3. 5 has two pairs and 1 leftover item. 7 has three pairs and 1 leftover item. When added together, the two leftover items join to form another complete pair. Since all items are now in pairs (6 pairs total = 12), the sum is even.
  4. Yes, Carlos is correct. An even number has all items paired up. Adding 1 introduces a single item with no partner, which creates an odd number.
  5. With an even number of socks, every sock will have a partner with none left over. If you have 17 socks, 17 is an odd number (ending in 7), so you will have 8 complete pairs and 1 sock left over without a partner.