Even and Odd Numbers: Definitions and Rules

What Are Even and Odd Numbers?

Even and odd numbers classify integers based on their divisibility by $2$. This mathematical classification is called parity.

An even number can be divided into two perfectly equal groups without any leftovers. When an even number is divided by $2$, the remainder is always $0$.

An odd number cannot be divided into two equal integer groups. When an odd number is divided by $2$, there is always a remainder of $1$.

Every integer is either even or odd, but no integer can be both.

Key Ideas and Vocabulary

The easiest way to determine the parity of any integer is to look at its ones digit.

• Even numbers always end in $0$, $2$, $4$, $6$, or $8$.
• Odd numbers always end in $1$, $3$, $5$, $7$, or $9$.

This rule works for both positive and negative numbers. For example, $-34$ ends in a $4$, so it is even. The number $157$ ends in a $7$, so it is odd.

Parity applies to all integers. Zero is an even number because dividing $0$ by $2$ yields a quotient of $0$ with a remainder of $0$. It is the first of the non-negative whole numbers, while the positive counting or natural numbers alternate starting with $1$, which is odd.

Algebraically, mathematicians write the general form of an even number as $2n$, where $n$ is any integer. The general form of an odd number is $2n + 1$ or $2n - 1$.

Visual Explanation

Visualizing numbers as arrays of paired objects quickly reveals whether they are even or odd.

If a set of objects can be arranged into pairs with nothing left over, the total count is even. If exactly one object is left unpaired, the total count is odd.

On a number line, even and odd integers alternate perfectly. If you start at $0$ and jump by $2$ in either direction, you land exclusively on even numbers.

Rules for Parity Operations

When combining integers, predictable parity patterns emerge. These rules help check answers without completing full calculations.

Adding or subtracting two even numbers always produces an even result. This demonstrates the closure property of even numbers under addition and subtraction. Adding or subtracting two odd numbers also produces an even result, because the two remainders of $1$ combine to form a new complete pair.

Multiplying integers follows its own logic. An even number multiplied by any integer results in an even number. An odd number multiplied by another odd number is the only way to produce an odd product.

Worked Examples

Apply the definitions and rules to solve these common parity problems.

Example 1: Identifying even and odd digits

Question: Which of the following numbers are even? $45$, $108$, $9{,}453$, $0$, $-16$

Method:

1. Look only at the ones digit of each number.
2. If the digit is $0$, $2$, $4$, $6$, or $8$, the number is even; if it is $1$, $3$, $5$, $7$, or $9$, it is odd.
3. The number $45$ ends in $5$, so it is odd.
4. The number $108$ ends in $8$, so it is even.
5. The number $9{,}453$ ends in $3$, so it is odd.
6. The number $0$ is perfectly divisible by $2$, so it is even.
7. The number $-16$ ends in $6$, so it is even.

Answer: The even numbers are $108$, $0$, and $-16$.

Check: Divide each even number by $2$. The results ($54$, $0$, and $-8$) are integers with no remainders.

Example 2: Determining the parity of a sum

Question: Without calculating the exact sum, is $347 + 591$ even or odd?

Method:

1. Identify the parity of the first number. The number $347$ ends in $7$, so it is odd.
2. Identify the parity of the second number. The number $591$ ends in $1$, so it is odd.
3. Apply the parity rules for addition. The sum of two odd numbers is always an even number.

Answer: The sum is even.

Check: Calculate the exact sum: $347 + 591 = 938$. The number $938$ ends in $8$, confirming it is even.

Example 3: Consecutive even numbers

Question: The sum of three consecutive even numbers is $42$. What are the three numbers?

Method:

1. Use the general algebraic form for even numbers, setting the first number as $2n$.
2. Write the next two consecutive even numbers as $2n + 2$ and $2n + 4$.
3. Set up the equation for their sum: $2n + (2n + 2) + (2n + 4) = 42$.
4. Combine the like terms to simplify the equation: $6n + 6 = 42$.
5. Subtract $6$ from both sides to isolate the variable term: $6n = 36$.
6. Divide by $6$ to solve for $n$, which gives $n = 6$.
7. Substitute $n = 6$ back into the original expressions: $2(6) = 12$, $2(6) + 2 = 14$, and $2(6) + 4 = 16$.

Answer: The three numbers are $12$, $14$, and $16$.

Check: Verify that they are consecutive even numbers and add them together: $12 + 14 + 16 = 42$. The solution is correct.

Common Mistakes and Non-Examples

A frequent misconception is assuming that parity applies to all types of numbers. Even and odd classifications apply only to integers.

Fractions, such as $\dfrac{1}{2}$, and decimals are non-examples of parity. For instance, $3.5$ is neither even nor odd. Although it ends in a $5$, the parity rule for the last digit only applies to whole integers, not to tenths or hundredths.

Another common mistake is claiming that negative numbers cannot be even or odd. Parity is symmetric around zero. The number $-8$ is even because $-8 \div 2 = -4$ with a remainder of $0$.

Real-World Connections

Even and odd numbers organize daily tasks. For instance, creating equal sports teams without leftover players requires an even number of participants.

Digital systems and computer programming rely heavily on parity. A "parity bit" is an extra binary digit added to data to ensure the total number of $1$-bits is either consistently even or odd. This acts as an automated answer-check method to detect transmission errors.

Street addresses also use parity. In many cities, houses on one side of a street are assigned even numbers, while houses on the opposite side receive odd numbers, making navigation logical and predictable.