Even and Odd Numbers: Rules, Properties and Examples

Even and Odd Numbers classify integers by divisibility by 2. An even integer has remainder 0 and an odd integer has remainder 1 when divided by 2. This page covers last-digit tests, parity rules for operations, algebraic expressions, consecutive integers, and counting even or odd values.

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Overview Even and Odd Numbers Formula & Tricks Even and Odd Numbers Concepts Even and Odd Numbers Video Lessons Practice Even and Odd Numbers Questions Even and Odd Numbers Quick Quiz Even and Odd Numbers Revision Notes Even and Odd Numbers FAQs

What Are Even and Odd Numbers?

An even number is an integer divisible by 2, while an odd number leaves remainder 1 when divided by 2. Every integer is either even or odd.

An even integer can be written as 2k and an odd integer can be written as 2k + 1, where k is any integer. Zero is even because 0 = 2 × 0. Negative integers also follow the same rule; for example, −8 is even and −7 is odd.

Even and Odd Numbers Formula & Tricks

Important Formulas

General form of an even number

Even number = 2k

Here, k is any integer. Every multiple of 2 is even.

General form of an odd number

Odd number = 2k + 1

Here, k is any integer. Division by 2 gives remainder 1.

Number of integers in a range

Number of integers from a to b = b − a + 1

Use this before counting even or odd integers in an inclusive range.

Product of consecutive integers

n(n + 1) is always even

Among two consecutive integers, one is always even.

Quick Tricks

Check only the last digit

For a whole number written in base 10, its parity depends only on its units digit. Digits 0, 2, 4, 6 and 8 indicate an even number; digits 1, 3, 5, 7 and 9 indicate an odd number.

Example: 78,432 is even because its last digit is 2, while 91,275 is odd because its last digit is 5.

Replace numbers by parity values

For a large expression, treat an even number as 0 and an odd number as 1. Apply the operation rules to determine only whether the result is even or odd.

Example: The product 248 × 735 is even because it contains an even factor, so the actual size of the product is not needed.

Use the nearest consecutive pair

A product containing two consecutive integers is automatically even because one member of every consecutive pair is even.

Example: The expression 101 × 102 is even because 102 is even.

Even and Odd Numbers Concepts

Identifying Even and Odd Numbers

A number is even if its units digit is 0, 2, 4, 6 or 8, and odd if its units digit is 1, 3, 5, 7 or 9.

This last-digit rule applies to positive and negative integers when the sign is ignored. It does not classify non-integers such as 3.5 as even or odd because evenness and oddness are defined for integers.

Example: −46 is even because its last digit is 6, whereas −39 is odd because its last digit is 9.

Parity Rules for Addition and Subtraction

The sum or difference of two numbers with the same parity is even, while the sum or difference of numbers with different parity is odd.

The rules are: even ± even = even, odd ± odd = even, even ± odd = odd, and odd ± even = odd. These rules apply repeatedly to expressions with several integer terms.

Example: 27 + 14 − 8 = 33, which is odd because odd + even − even remains odd.

Parity Rules for Multiplication

A product is even if at least one factor is even; a product is odd only when every factor is odd.

The rules are: even × even = even, even × odd = even, and odd × odd = odd. Therefore, one even factor is sufficient to make the entire integer product even.

Example: 13 × 17 is odd, but 13 × 16 × 17 is even because the factor 16 is even.

Powers and Algebraic Expressions

A positive power of an even integer is even, and a positive power of an odd integer is odd.

For an integer n, n², n³ and all higher positive powers have the same parity as n. Also, n(n + 1) is always even because n and n + 1 are consecutive. The expression n² + n is therefore always even.

Example: If n = 7, then n² + n = 49 + 7 = 56, which is even.

Counting Even and Odd Integers in a Range

In a sequence of consecutive integers, even and odd numbers alternate. An even-length sequence contains equal numbers of evens and odds.

For an inclusive range from a to b, first calculate the total count b − a + 1. If the count is even, each parity occurs half as many times. If the count is odd, the parity of the starting and ending numbers determines which type occurs once more.

Example: From 12 to 20 there are 9 integers: five even numbers (12, 14, 16, 18, 20) and four odd numbers.