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Algebraic Proof

How to prove statements about integers, positivity and algebraic identities, and use a counterexample to disprove a claim.

Explain why the claim holds in every case

A proof uses algebra and clear reasons to cover every value allowed by the statement. Checking several examples can suggest a pattern but cannot prove a universal claim. One valid counterexample is enough to disprove it.

For an integer n, write an even integer as 2n and an odd integer as 2n+1. Consecutive integers differ by 1; consecutive even or odd integers differ by 2

Worked example 1

Prove that the sum of three consecutive even integers is divisible by 6.

  1. Write the integers as 2n, 2n+2 and 2n+4, where n is an integer. Their sum is 6n+6

  2. Your turn. Complete 6n+6=6(\dots). What goes in the bracket?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    6n+6=6(n+1)

  3. Answer: the sum is 6(n+1). Since n+1 is an integer, the sum is divisible by 6

Worked example 2

Prove that the product of two consecutive odd integers leaves remainder 3 when divided by 4.

  1. Write them as 2n+1 and 2n+3. Multiply: (2n+1)(2n+3)=4n^2+8n+3

  2. Your turn. Factorise 4n^2+8n by taking out 4. What remains inside the bracket?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    4n^2+8n=4(n^2+2n)

  3. Answer: the product is 4(n^2+2n)+3. The bracket is an integer, so this is a multiple of 4 plus remainder 3

Worked example 3

Prove that (n+4)^2-(n-2)^2 is divisible by 12 for every integer n

  1. Use a difference of squares: \bigl((n+4)-(n-2)\bigr)\bigl((n+4)+(n-2)\bigr)

  2. Your turn. Simplify (n+4)-(n-2)

    The difference is 6

  3. The sum is 2n+2, so the expression is 6(2n+2)=12(n+1)
  4. Answer: 12(n+1) is divisible by 12 because n+1 is an integer.

Connect the final form to the statement

An expanded expression is not always a finished proof. To prove a multiple, factor out the required number and explain that the other factor is an integer. To prove positivity, rewrite the expression using a square plus a positive amount.

Worked example 4

Prove that x^2-8x+19>0 for every real x

  1. Complete the square: x^2-8x+19=(x-4)^2+3

  2. Your turn. Because (x-4)^2\ge0, what is the smallest possible value of the expression?

    The minimum value is 3, at x=4

  3. Answer: (x-4)^2+3\ge3>0, proving the claim for all real x

Worked example 5

Prove that \frac{x^2-4}{x+2}-\frac{x^2-9}{x+3}=1 whenever both fractions are defined.

  1. The original denominators require x\ne-2,-3. Factorise and cancel on this domain: \frac{(x-2)(x+2)}{x+2}=x-2 and \frac{(x-3)(x+3)}{x+3}=x-3

  2. Your turn. Simplify (x-2)-(x-3)

    x-2-x+3=1

  3. Answer: the identity holds for every x\ne-2,-3. The excluded inputs remain excluded after cancellation.

Worked example 6

Someone claims that n^2+n+17 is prime for every non-negative integer n. Disprove the claim.

  1. Try n=16, which is an allowed non-negative integer.

  2. Your turn. Calculate 16^2+16+17

    The value is 289

  3. 289=17^2, so it has factors other than 1 and itself.
  4. Answer: the claim is false; n=16 is a counterexample. One counterexample is enough, even if many smaller inputs produce primes.

Common mistakes

  • Using examples as proof. Use algebra to show the result holds for every allowed value.
  • Putting consecutive odd integers one apart. Use 2n+1 and 2n+3, where n is an integer.
  • Losing signs when subtracting. Bracket the entire expression being subtracted.
  • Claiming a multiple without justification. Explain why the remaining factor is an integer.
  • Ignoring denominator restrictions. Exclude values that make an original denominator zero.

Now try it: Algebraic Proof practice questions

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