An odd number leaves a remainder of 1 when divided by 2. Examples include 1, 3, 7, 15, and −9. The last digit is 1, 3, 5, 7, or 9. Odd numbers cannot be split into two equal whole groups.

Parity—being odd or even—has simple rules. Odd + odd = even. Odd + even = odd. Odd × odd = odd. These rules help with mental arithmetic and with proofs. In number theory, parity is a basic tool.

Odd numbers appear in patterns, sequences, and puzzles. The sum of the first n odd numbers is n². That identity has a neat geometric proof using squares. In computer science, the least significant bit indicates parity, which is used in error detection and algorithms.

• Integer not divisible by two
• Last digit 1, 3, 5, 7, or 9
• Parity rules for addition and multiplication
• Sum of first n odd numbers is n²