Odd integers
Also known as: odd numbers
Odd integers are integers that are not divisible by 2, such as −3, −1, 1, 3, and 5. Every odd integer can be written in the form 2n + 1, where n is an integer.
An odd integer is any integer that cannot be divided evenly by 2 — dividing it by 2 always leaves a remainder of 1. The odd integers are ..., −5, −3, −1, 1, 3, 5, ..., and every one of them can be expressed algebraically as 2n + 1 for some integer n. Integers that are divisible by 2 (including zero) are even.
Odd and even integers follow reliable arithmetic rules worth memorizing. For addition and subtraction: odd + odd = even, even + even = even, and odd + even = odd. For multiplication: a product is odd only when every factor is odd — odd × odd = odd, while anything multiplied by an even number is even. So 7 × 9 = 63 is odd, but 7 × 8 = 56 is even because a single even factor forces an even result.
These parity rules turn intimidating problems into quick logic checks. Asked whether 3a + 2b is odd when a and b are integers with a odd, you can reason it out: 3a is odd × odd = odd, 2b is always even, and odd + even = odd — no plugging in required. Consecutive odd integers, which differ by 2 (such as n, n + 2, n + 4), are another common setup in word problems.
Odd and even number properties appear on the quantitative sections of the GRE and the math sections of the ASVAB. Test writers favor questions that ask which expressions must be even or odd, so knowing the parity rules cold is faster and safer than testing sample numbers under time pressure.
Key takeaways
- Odd integers leave a remainder of 1 when divided by 2 and take the form 2n + 1.
- Odd + odd = even, odd + even = odd, and even + even = even.
- A product is odd only if every one of its factors is odd.
- Consecutive odd integers differ by 2, as in n, n + 2, n + 4.
- The GRE and ASVAB test parity rules through must-be-even or must-be-odd questions.