Proof and counterexamples

The idea

Proof by deduction uses general algebra: an even number is $2n$ , an odd number $2n + 1$ . Proof by exhaustion checks every case of a finite list. One counterexample is enough to disprove a statement.

Write the numbers in general form, or list every case.
Simplify into a form that shows the claim, such as $3(\dots)$ for a multiple of $3$ .
Finish with a sentence that states what has been shown.

Watch out Checking a few examples does not prove a statement about all numbers.

See it worked

Worked example

Prove that the sum of any three consecutive integers is a multiple of $3$ .

Let the integers be $n$ , $n + 1$ and $n + 2$ .
Their sum is $3n + 3 = 3(n + 1)$ .
$n + 1$ is an integer, so the sum is a multiple of $3$ .

Answer The sum is $3(n + 1)$ , a multiple of $3$

Your turn

Question 1Calculator allowed3 marks

Find the smallest positive integer $n$ that shows the claim ' $n^2 + n + 41$ is prime for every positive integer $n$ ' is false.

Question 2Non-calculator4 marks

Prove by exhaustion that a square number leaves remainder $0$ or $1$ when divided by $4$ .

Write your answer, then show the working

Every integer is $2k$ or $2k + 1$ .
$(2k)^2 = 4k^2$ : remainder $0$ .
$(2k + 1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1$ : remainder $1$ .
Both cases are covered, so the claim holds.

Answer Every integer is even or odd: $(2k)^2 = 4k^2$ and $(2k + 1)^2 = 4(k^2 + k) + 1$

Question 3Non-calculator3 marks

Find the smallest positive integer $n$ for which $2^n + 1$ is not prime.

In past papers

38 official questions in 34 papers since 2018. The latest:

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