EQN-01-11

Even, Odd & Consecutive Integers

Use parity rules and the algebra of consecutive integers to solve number-property puzzles fast.

~5 min read•Difficulty 2/5

even \pm even = even odd \pm odd = even even \pm odd = odd

even \times anything = even odd \times odd = odd

A product is even if any factor is even; it is odd only when all factors are odd.

even number : 2 n

odd number : 2 n + 1

consecutive : n, n + 1, n + 2

consecutive even : n, n + 2, n + 4 (n even)

consecutive odd : n, n + 2, n + 4 (n odd)

The sum of two consecutive integers is always odd.
The sum of three consecutive integers is 3 × the middle one (so divisible by 3).

7 + 8 + 9 = 24 = 3 \times 8

Parity often eliminates answer choices instantly: if a result must be even, cross out the odd options before computing.

Worked examples

Example 1 (difficulty 2/5)

If n is an odd integer, which of these is always even?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

The sum of three consecutive integers is 72. What is the largest?

◎ Predict the answer first:

Writing consecutive even integers as

n, n + 1, n + 2

Why: It mirrors plain consecutive integers.

Fix: Even (or odd) consecutives step by

2 : n, n + 2, n + 4.

Assuming a product of odds could be even

Why: Large products "feel" even.

Fix: Odd

\times

odd is always odd; you need an even factor to get an even product.