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The Fundamental Theorem of Arithmetic sheet
3-mark answer frame

Show that 12ⁿ cannot end with the digit 0 or 5 for any natural number n.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. Given: n is a natural number. To show: 12ⁿ does not end with 0 or 5.
  2. 2 · Write the theorem. A number ending in 0 or 5 is divisible by 5. By the Fundamental Theorem of Arithmetic, its prime factorisation must then contain the prime 5.
  3. 3 · Factorise. 12 = 2² × 3, so 12ⁿ = (2² × 3)ⁿ = 2²ⁿ × 3ⁿ.
  4. 4 · Compare. The only primes in 12ⁿ are 2 and 3. Prime factorisation is unique, so 5 can never appear.
  5. 5 · Check and box the answer. Therefore 12ⁿ is never divisible by 5, so it cannot end with 0 or 5 for any natural number n.

Scoring tip: Always quote the uniqueness part of the theorem. It is the reason the prime 5 cannot turn up later.