Even and odd numbers are among the first patterns children notice in arithmetic, and they remain useful far beyond school, from checking calculations to writing proofs. This guide covers the definitions, the rules and the reasoning behind them.
Definitions
Even number: an integer that can be written as 2k, where k is an integer.
Odd number: an integer that can be written as 2k + 1, where k is an integer.
Informally, a number is even if that many objects can be put into pairs with none left over, and odd if exactly one object is left without a partner. Whether a number is even or odd is called its parity.
- Even numbers: … −6, −4, −2, 0, 2, 4, 6, 8 …
- Odd numbers: … −5, −3, −1, 1, 3, 5, 7, 9 …
How to tell if a number is even or odd
- Look at the last digit. 0, 2, 4, 6 or 8 means even; 1, 3, 5, 7 or 9 means odd.
- Divide by 2. Remainder 0 means even, remainder 1 means odd.
- Pair objects. Useful with young learners: line up counters in pairs and see if one is left over.
You can try any number with the Even and Odd Test, which shows the division step by step.
Rules for adding, subtracting and multiplying
| Operation | even ○ even | even ○ odd | odd ○ odd |
|---|---|---|---|
| Addition (+) | even | odd | even |
| Subtraction (−) | even | odd | even |
| Multiplication (×) | even | even | odd |
Try the rules with your own numbers:
Parity rules explorer
Odd result
6 + 7 = 13, which is odd.
Rule: even + odd = odd.
Why the rules hold
Write the odd numbers as 2a + 1 and 2b + 1. Their sum is 2a + 2b + 2 = 2(a + b + 1), a multiple of 2, so odd + odd = even. Their product is 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which has the odd form, so odd × odd = odd. Any product with an even factor 2a is 2 × (something), so it is even.
Division and parity
Division has no simple parity rule. 12 ÷ 2 = 6 (even ÷ even = even), 12 ÷ 4 = 3 (even ÷ even = odd) and 10 ÷ 4 = 2.5 (not an integer at all). Check each case individually.
Special cases
- Zero is even, because 0 = 2 × 0. See Is Zero Even or Odd? for four separate arguments.
- Negative numbers have parity just like positive ones: −12 is even and −11 is odd.
- Decimals and fractions such as 3.5 or ½ are neither even nor odd; parity applies only to integers.
- 2 is the only even prime. Every other even number is divisible by 2 and so is composite.
Even and odd numbers in everyday life
Parity quietly organises many systems. House numbers commonly put evens on one side of a street. Many “check digit” schemes, such as the Luhn algorithm used on bank cards, double alternate digits, a rule built on positions being even or odd. In computing, the lowest bit of a binary number is its parity, which is why checking n & 1 is a common programming idiom.
Practice questions
- Is 3,457 even or odd? (Odd: last digit 7.)
- What is the parity of 15 + 27 + 8? (Even: two odd terms make an even sum, plus an even term.)
- Is 7 × 9 × 11 even or odd? (Odd: all factors are odd.)
- Is −100 even? (Yes: −100 = 2 × (−50).)
Frequently asked questions
What is the parity of a number?
Parity is the property of being even or odd. Two numbers have the same parity if both are even or both are odd.
Is every integer either even or odd?
Yes, every integer is exactly one of the two. Dividing by 2 always leaves remainder 0 or 1, never both.
Are even and odd numbers taught with negative numbers?
Early lessons focus on positive whole numbers, but the definitions work for all integers, and later courses use negative examples.
Is infinity even or odd?
Neither. Infinity is not an integer, so parity does not apply.