What is an even number in math?
An even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.
Even Number in a picture
Formal definition and notation
An integer m is even when m = 2n for some integer n, that is, when 2 is a factor of m and m ÷ 2 leaves remainder 0. Integers that are not even are odd and have the form 2n + 1. The even numbers are …, −4, −2, 0, 2, 4, …; fractions and decimals such as 3.4 are neither even nor odd.
| Symbol | Read it as | Meaning |
|---|---|---|
| 2n | two n | the general even number, where n is any integer |
| 2n + 1 | two n plus one | the general odd number, one more than an even number |
Key facts
- Last-digit test: a whole number is even exactly when its ones digit is 0, 2, 4, 6 or 8. Only the last digit matters: 9,999,994 is even.
- Zero is even: 0 = 2 × 0, and 0 sits between the odd numbers −1 and 1.
- Even and odd numbers alternate on the number line: even, odd, even, odd, …
- 2 is the only even prime number. Every other even number has 2 as a factor, so it is composite.
- Half of an even number is a whole number: 58 ÷ 2 = 29. That is the pairing idea in numbers.
Even and odd rules
- even × any integer
- always even: 6 × 7 = 42
- odd × odd
- always odd: 5 × 9 = 45
Why: 2a + 2b = 2(a + b), and (2a + 1) + (2b + 1) = 2(a + b + 1). Both are 2 times an integer.
Even Number examples, step by step
Example 1: Use the last-digit test
Is 3,578 even or odd?
- Look only at the ones digit: 8.
- 8 is one of 0, 2, 4, 6, 8, so the number is even.
- Check: 3578 ÷ 2 = 1789 with no remainder.
Answer: 3,578 is even.
Example 2: Even or odd without calculating
Is 47 × 18 + 33 even or odd? Decide without working out the total.
- 47 × 18: odd times even is even.
- Then even + 33: even plus odd is odd.
- Check: 47 × 18 = 846 and 846 + 33 = 879, which ends in 9.
Answer: 47 × 18 + 33 is odd (it equals 879).
Example 3: Count the even numbers in a range
How many even numbers are greater than 15 and less than 75?
- The first even number after 15 is 16; the last before 75 is 74.
- The evens are 2 × 8, 2 × 9, …, 2 × 37.
- Count the multipliers from 8 to 37: 37 − 8 + 1 = 30.
Answer: There are 30 even numbers between 15 and 75.
Common mistakes
The definition only asks whether dividing by 2 leaves remainder 0. It does for 0, and 0 fits the even, odd, even pattern between −1 and 1.
Only the ones digit decides. The other digits stand for tens, hundreds and so on, which are always even amounts.
Even and odd apply only to integers. The last-digit test works only for whole numbers and their negatives.
Not to be confused with
An even number divides by 2 with no remainder (2n); an odd number leaves remainder 1 (2n + 1). 14 is even and 15 is odd.
Even is about dividing by 2; prime means exactly two factors. 2 is both even and prime, but 10 is even and not prime, and 7 is prime and not even.
Even numbers are exactly the multiples of 2. A multiple of another number, such as 15 (a multiple of 5), can be odd.
Quick check: 3 questions
Which number is even?
Answer: B. 4,092 ends in 2, so it is even. The others end in 7, 1 and 5, which are odd digits. Starting with an even digit (like 2,461 or 8,005) does not matter.Show answer and explanation
If n is any integer, which expression is always even?
Answer: D. 2n + 6 = 2(n + 3), which is 2 times an integer, so it is always even. n + 2 and 3n are odd when n is odd, and 2n + 1 is always odd.Show answer and explanation
What is the sum of an odd number and an odd number?
Answer: B. (2a + 1) + (2b + 1) = 2(a + b + 1), which is even. For example, 7 + 9 = 16. The two leftover ones pair up with each other.Show answer and explanation
Even Number: frequently asked questions
Is 0 an even number?
Yes. A number is even when dividing it by 2 leaves remainder 0, and 0 ÷ 2 = 0 with remainder 0. You can also write 0 = 2 × 0. On the number line, 0 sits between the odd numbers −1 and 1, which fits the even, odd, even pattern. Zero is the only number that is neither positive nor negative, but it is definitely even.
Can negative numbers be even?
Yes. −2, −4, −36 and −100 are all even, because each equals 2 times an integer: −36 = 2 × (−18). Negative numbers such as −3 and −15 are odd. Only fractions and decimals that are not integers, such as 12 or 4.6, are neither even nor odd.
How can you tell if a big number is even?
Look at the last digit only. If it is 0, 2, 4, 6 or 8, the number is even, no matter how many digits come before it. So 7,345,918 is even and 2,000,001 is odd. This works because every place except the ones place stands for a multiple of 10, and 10 is even.
Practice and tools
Sources
- Common Core State Standards, Grade 2 Operations and Algebraic Thinking (2.OA.C.3)
- OpenStax, Prealgebra 2e, 2.4 Find Multiples and Factors (divisibility by 2)
- Weisstein, Eric W. "Even Number." MathWorld, Wolfram Research
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 2.OA.C.3
Cite this entry
Effortless Math Team. “Even Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/even-number/
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