Prime factorisation shows the structure of a number: a number is a perfect square exactly when every prime in its factorisation has an even power. To prove a statement about odd, even or consecutive numbers, write them algebraically, such as 2n for an even number and 2n + 1 for an odd number, so the argument covers every case. To show a statement is false, one counterexample is enough.
Tools for number proof
- Even number: 2n. Odd number: 2n + 1
- Consecutive integers: n, n + 1, n + 2
- Two different odd numbers: 2a + 1 and 2b + 1
- Perfect square: every prime has an even power
- A multiple of k: k × (an integer)
Perfect squares from prime factors
Write the number as a product of primes. If every power is even, the number is a perfect square, because each prime can be split equally between the two identical factors. 200 = 2³ × 5² is not a perfect square, because the power of 2 is odd. Multiplying by one more 2 would make it 400 = 2⁴ × 5², which is 20².
Writing numbers algebraically
To prove something is always true, examples are not enough; you need algebra that covers every case. Any even number can be written as 2n and any odd number as 2n + 1, where n is an integer. For two different odd numbers, use different letters, 2a + 1 and 2b + 1, otherwise you have only proved the result for two equal numbers.
Structure of a proof
Write the expression for the situation, simplify it, then rearrange it into a form that shows the result: 2 × (an integer) for an even number, 2 × (an integer) + 1 for an odd number, or k × (an integer) for a multiple of k. Finish with a sentence stating what you have shown.
Counterexamples
To disprove a statement, one example where it fails is enough. “Every prime number is odd” is false, because 2 is prime and even. Some statements look true for many cases and fail later: n² + n + 41 is prime for n = 0 to 39, but when n = 40 it equals 1681 = 41².
Consecutive numbers
Consecutive integers are n, n + 1, n + 2 and so on. Their sum and products often have neat properties. For example, the difference between the squares of two consecutive integers is (n + 1)² − n² = 2n + 1, which is always odd.
Worked examples
Example 1. Is 200 a perfect square? Use its prime factorisation to justify your answer.
- 200 = 2³ × 5².
- The power of 2 is odd.
Answer: No, because 2 appears an odd number of times.
Example 2. 288 = 2⁵ × 3². Find the smallest whole number k such that 288k is a perfect square.
- The power of 3 is already even; the power of 2 is odd.
- Multiply by one more 2 to make 2⁶ × 3², which is (2³ × 3)² = 24².
Answer: k = 2, giving 576 = 24².
Example 3. Prove that the sum of two odd numbers is always even.
- Let the odd numbers be 2a + 1 and 2b + 1.
- (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1).
Answer: 2(a + b + 1) is a multiple of 2, so the sum is always even.
Example 4. Prove that the product of two odd numbers is always odd.
- (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1.
- = 2(2ab + a + b) + 1.
Answer: This is 2 × (an integer) + 1, so the product is odd.
Example 5. Show that the statement “n² + n + 41 is always prime” is false.
- Try n = 40: 40² + 40 + 41 = 1600 + 81 = 1681.
- 1681 = 41 × 41.
Answer: n = 40 is a counterexample, because 1681 is not prime.
Common mistakes
- Proving with examples. Testing a few numbers does not prove a result for all numbers; use algebra.
- Using the same letter for two different odd numbers. 2n + 1 and 2n + 1 are equal; use 2a + 1 and 2b + 1.
- Not finishing the proof. Write the result in a form such as 2(…) and state the conclusion.
- Forgetting that 2 is prime. 2 is the only even prime, which makes it a useful counterexample.
- Not using prime factors for perfect squares. Check whether every prime has an even power.
How number proof is examined
Algebraic proof and prime factorisation problems are Higher tier content in Edexcel GCSE (1MA1); the International GCSE (4MA1) includes similar reasoning questions. Typical tasks:
- use a prime factorisation to find the smallest multiplier that makes a perfect square or cube
- prove a result about odd, even or consecutive numbers
- show that a statement is false with a counterexample
- explain why a number with a given factorisation is or is not a square
Practise proofs with the algebraic proof worksheet, which has full solutions, and prime factors with the GCSE number revision engine.
Prime factorisation itself is covered in HCF and LCM using prime factors; proofs about expressions are in algebraic proof.
Frequently asked questions
How do you write an odd number in algebra?
As 2n + 1, where n is an integer.
How do you prove something is always even?
Write it algebraically and rearrange it into the form 2 × (an integer).
What is a counterexample?
A single example that shows a statement is false.