Proof and disproof

Proof by deduction

Start from known facts and use clear logical steps. Write general numbers algebraically: an even number is 2n, an odd number is 2n + 1, consecutive integers are n and n + 1 (n an integer).

Example: (2m + 1) + (2n + 1) = 2(m + n + 1), which is even.

Proof by exhaustion

Split the situation into a finite number of cases and check every one. For example, to prove something about all integers, you might test n = 3k, 3k + 1 and 3k + 2, which covers every integer.

Disproof by counter-example

To show a statement is false, find one example where it fails. Checking lots of examples that work never proves a general statement.

Worked example

Prove that the square of any odd number is one more than a multiple of 4.

1. Let the odd number be 2n + 1, where n is an integer.
2. (2n + 1)² = 4n² + 4n + 1
3. = 4(n² + n) + 1, and n² + n is an integer

Answer: So the square is one more than a multiple of 4.

Check yourself

1. Which value of n is a counter-example to 'n² + n + 41 is prime for every positive integer n'?
   • n = 1
   • n = 41
   • n = 2
   • n = 10
   Show answer

   n = 41. 41² + 41 + 41 = 41 x 43, which is not prime.

2. Which statement is shown to be false by x = -3?
   • If x > 2 then x² > 4
   • x² ≥ 0 for all real x
   • If x < 0 then x³ < 0
   • If x² > 4 then x > 2
   Show answer

   If x² > 4 then x > 2. x = -3 gives x² = 9 > 4, but -3 is not greater than 2.

3. A student checks a statement for 100 different values of n and it works every time. What has been shown?
   • The statement is proved for all n
   • Nothing general: it is not a proof for all n
   • The statement is proved by exhaustion
   • The statement is false
   Show answer

   Nothing general: it is not a proof for all n. Examples never prove a statement about infinitely many values.

4. To prove by exhaustion that n² + 1 is never a multiple of 3, which cases cover every integer n?
   • n even and n odd
   • n = 1, n = 2 and n = 3
   • n = 3k, n = 3k + 1, n = 3k + 2
   • n = 3k only
   Show answer

   n = 3k, n = 3k + 1, n = 3k + 2. Every integer leaves remainder 0, 1 or 2 when divided by 3.

5. Using (2n + 1)² = 4(n² + n) + 1, what is the remainder when the square of any odd number is divided by 4?
   Show answer

   1. 4(n² + n) is a multiple of 4, so the remainder is 1.

6. If n is an integer, which expression is always even?
   • n + 2
   • n²
   • 2n
   • 2n + 1
   Show answer

   2n. 2n is a multiple of 2 for every integer n.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.