Algebraic Proof Toolkit for Edexcel International GCSE (Higher): Standard Forms That Make Proofs Easy

Proving that the sum of three consecutive integers is divisible by 3.
Proving that the sum of three consecutive integers is divisible by 3.

Algebraic proof questions in Edexcel International GCSE (Higher) often look difficult because they are written in words. The quickest way to handle them is to translate the words into standard algebraic forms that guarantee the number property you need (even, odd, multiple, consecutive, square, etc.). Once the translation is correct, the rest of the proof is usually straightforward simplification, factoring, and a clear final statement.

This post gives a compact “toolkit” of the most common forms, presented in a table you can reuse, plus a small set of extras and techniques that frequently appear in Higher-tier proof questions.

The core principle

In an algebraic proof, represent the numbers so the required property is built in. For example:

  • If a number is even, write it as 2n for some integer n.
  • If a number is a multiple of 3, write it as 3n for some integer n.

The phrase “for some integer n” matters. It ensures the representation really matches the number fact.

Standard representations table

Number fact Written using algebra
Even Number 2n
Odd Number 2n+1 or 2n-1
Multiple of 3 3n
Consecutive Numbers n, n+1, n+2, ...
Consecutive Even Numbers 2n, 2n+2, 2n+4, ...
Consecutive Odd Numbers 2n+1, 2n+3, 2n+5, ...
Consecutive Square Numbers n², (n+1)², (n+2)², ...

Useful extras for Higher-tier proofs

Multiples of any integer k

If a number is a multiple of k, write it as:

  • kn

Examples: multiple of 4 is 4n, multiple of 7 is 7n.

Numbers that leave a remainder (common “mod” patterns)

These forms appear in “show that” questions about divisibility and remainders.

  • Remainder 1 when divided by 3: 3n + 1
  • Remainder 2 when divided by 3: 3n + 2
  • Remainder 1 when divided by 5: 5n + 1

Consecutive multiples of k

Consecutive multiples of k differ by k:

  • kn, k(n+1), k(n+2), ...

Two or three consecutive integers

  • Two consecutive integers: n and n+1
  • Three consecutive integers: n, n+1, n+2

Square expansions to know

  • (n+1)² = n² + 2n + 1
  • (n+2)² = n² + 4n + 4

Techniques that prove divisibility quickly

Technique 1: Factor out what you want to prove

To prove an expression is divisible by 3, aim to rewrite it in the form 3(something). Similarly, for divisibility by 5, aim for 5(something), and so on.

Technique 2: Use parity forms early

If the conclusion involves even or odd, substitute 2n or 2n+1 early. The final expression usually becomes:

  • 2(something) → even
  • 2(something)+1 → odd

Technique 3: Keep the “consecutive” structure

Consecutive numbers must stay linked. Write them as n, n+1, n+2 rather than using unrelated letters. This preserves the one-step difference that makes the proof work.

Mini worked examples (Higher style)

Example A: Prove the sum of two consecutive integers is odd

Let the consecutive integers be n and n+1.

Sum:

n + (n+1) = 2n + 1

Since 2n+1 is of the form 2k+1, the sum is odd.

Example B: Prove the sum of three consecutive integers is divisible by 3

Let the integers be n, n+1, n+2.

Sum:

n + (n+1) + (n+2) = 3n + 3 = 3(n+1)

This is divisible by 3 because it has a factor of 3.

Example C: Prove the difference between consecutive squares is odd

Consider consecutive squares (n+1)² and n².

Difference:

(n+1)² − n² = (n² + 2n + 1) − n² = 2n + 1

2n+1 is odd, so the difference between consecutive squares is always odd.

Common mistakes that lose marks

  • Not stating n is an integer. Proofs depend on integer properties.
  • Breaking “consecutive”. If numbers are consecutive, they must be written as n, n+1, n+2 (not a, b, c with no links).
  • Stopping too early. After simplifying, explicitly connect the final form to the conclusion (for example, “= 3(n+1), so it is divisible by 3”).
  • Algebra slips. Proof questions are often easy conceptually, but only if expansion and factoring are accurate.

A quick checklist for algebraic proofs

  • Identify the number facts (even/odd/multiple/consecutive/square).
  • Translate using the standard forms (table + extras).
  • Simplify carefully (expand, collect like terms).
  • Factor to expose divisibility or parity.
  • Finish with a clear final sentence that states what the algebra proves.

With these standard forms and techniques, most Edexcel International GCSE (Higher) algebraic proof questions become a consistent process: translate correctly, simplify cleanly, and finish with a clear conclusion.

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