Even and Odd Numbers
Every integer is either even or odd.
Formula
Variables
Example
10 = 2·5 is even; 11 = 2·5 + 1 is odd.
How it works
An integer is even if it can be written as 2 times another integer, because that means it is divisible by 2 with no remainder. An integer is odd if it is exactly one more than an even number, so it leaves remainder 1 when divided by 2. This works because dividing any whole number by 2 can only produce remainder 0 or 1, so every integer fits one of these two forms.
When to use it
Use this when you need to classify integers as even or odd, such as checking parity in arithmetic, algebra, or number patterns. It is also helpful for reasoning about sums, products, and divisibility in basic proofs and problem-solving.
Worked example, step by step
Check whether 24 and 37 are even or odd. 24 = 2 × 12, so 24 fits the form n = 2k and is even. 37 = 2 × 18 + 1, so 37 fits the form n = 2k + 1 and is odd. Final answer: 24 is even and 37 is odd.
Common mistakes to avoid
- Thinking a number is even just because it ends in 0, 2, 4, 6, or 8 without checking that it is an integer
- Using 2k+1 for an even number or 2k for an odd number
- Forgetting that k must be an integer, not a decimal or fraction
Did You Know?
Parity — evenness or oddness — is the simplest error-detecting check and appears throughout computer science.
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