Checking examples is not a proof: the result must hold for every whole number. Use letters.

• An even number: 2n. An odd number: 2n + 1.
• Consecutive numbers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2.
• A multiple of 3: 3n.

Two different odd numbers need two letters: 2a + 1 and 2b + 1.

1. prove (n + 3)² − (n − 3)² is a multiple of 12
2. (n + 3)² = n² + 6n + 9
3. (n − 3)² = n² − 6n + 9
4. difference = 12n
5. 12n is a multiple of 12, for every n

Finish with a sentence that says why: "12n = 12 × n, so it is a multiple of 12".