Algebraic proof and counterexamples

An integer can be written as n, an even integer as 2n and an odd integer as 2n + 1, where n is an integer. Substitute these forms into the claim, simplify, and identify the property that proves the result. To prove divisibility by k, show the expression equals k times an integer.

Checking several numbers can suggest a pattern but cannot prove a statement for all integers. A single counterexample does disprove a universal claim: it must satisfy the assumptions and fail the conclusion. State the domain throughout the argument. Dividing by an expression which might be zero can silently exclude the very case needed for a proof.

Worked example

Prove that the product of two consecutive integers is even.

1. Write the integers as n and n + 1, where n is an integer.
2. One of two consecutive integers is even: if n is odd, n + 1 is even.
3. An even integer times any integer is even.

Answer: n(n + 1) is even for every integer n