Parity: odd and even numbers

What is parity?

Think of a number as a set of dots. If the dots can all be put into pairs, with none left alone, the number is even: 6 makes 3 pairs and 10 makes 5. If one dot is always left without a partner, the number is odd, like 7 or 11. So an odd number is a set of pairs plus one extra dot, or, looked at the other way, a set of pairs with one dot missing. Whether a number is odd or even is called its parity.

Adding even and odd numbers

The dot pictures act as proofs. Put two even numbers together and every dot still has a partner, so even + even = even, however many even numbers we add. Put two odd numbers together and the two leftover dots pair up with each other, so odd + odd = even. An even number and an odd number together keep exactly one leftover, so even + odd = odd. For longer sums, only the odd numbers matter: with 4 or 6 odd numbers the leftovers all pair up and the sum is even, while with 3 or 5 odd numbers one dot is left over and the sum is odd.

Puzzles that parity solves

Kishor must choose five number cards that add up to 30, but every card he has is odd: 13, 9, 1, 7, 11, 5 or 3. Checking every choice would take ages, yet parity settles it at once. Five odd numbers make an odd sum, and 30 is even, so the puzzle has no answer.

Martin and Maria’s birthdays are exactly a year apart. Can their ages add up to 112? Counting numbers take turns, odd, even, odd, even, so two consecutive numbers are always one odd and one even, and their sum is odd: 51 + 52 = 103. Since 112 is even, it is impossible.

Differences and products

Subtraction follows the same pattern as addition: even − even and odd − odd are even, while even − odd and odd − even are odd. Products behave differently. Think of a grid of small squares with some number of rows and columns. A 3 × 3 grid has 9 squares, odd, but a 3 × 4 grid has 12, even. If either side is even, the grid can be cut into pairs of squares, so the product is even. The product is odd only when both sides are odd. So 27 × 13 is odd, while 42 × 78 and 135 × 654 are even, all without multiplying.

Parity of expressions

An algebraic expression can have a fixed parity or a changing one. 3n + 4 is odd when n = 3 (13) but even when n = 8 (28). Expressions such as 100p, 2n or 48w − 2 are always even, because 2 divides every value; 2n + 1 and 4n + 3 are always odd. To list every even number we use 2n, for n = 1, 2, 3, …, and to list every odd number we use 2n − 1. The 100th even number is 200, and since each odd number is one less than the even number in the same position, the 100th odd number is 199.

Why it matters

Parity is one of the quickest tools in mathematics for proving that something is impossible. It also helps with magic squares and with the pattern of odd and even Virahanka–Fibonacci numbers. Answers to every question are in the NCERT solutions for Chapter 6; see also the full Chapter 6 notes.