--- ## Definition > [!tldr] Definition > An integer is said to be **even** if it is a multiple of 2. That is, the integer $n$ is even if there exists an integer $k$ such that $n = 2k$. > > An integer is said to be **odd** if it is one more than a multiple of 2. That is, the integer $n$ is odd if there exists another integer $k$ such that $n = 2k+1$. Notes: - We can also say that a number is even if it is *divisible* by 2 -- that is, $n$ is even if $n/2$ is an integer. - But we default to the language of "multiples" in the definition because they are often easier to think about and less prone to issues involving division. ## Examples and Non-Examples - The number $278$ is even because $278 = 2 \cdot 139$. That is, there exists another integer ($139$) such that $278$ is twice this. - The number $799$ is odd because $799 = 798 + 1$ and $798$ is even (since $798 = 2 \cdot 399$). That is, $799 = 2(399) + 1$. - The number $0$ is even since $0 = 2 \cdot 0$. - Negative numbers can also be even or odd. For example $-24$ is even since $-24 = 2 \cdot (-12)$. And $-19$ is odd since $-19 = 2(-10) + 1$. ## Resources