Prime Numbers and Factors

La méthode, étape par étape

1. 1

   Find factors in pairs

   24 → 1×24, 2×12, 3×8, 4×6 → 1, 2, 3, 4, 6, 8, 12, 24

   Working in pairs guarantees none are missed and tells you when to stop: once the pair members meet or cross, the list is complete. Listing factors randomly is how students end up one or two short, which then breaks every HCF question built on it.

2. 2

   Test primality by dividing by primes only

   Is 91 prime? 2 no, 3 no, 5 no, 7 → 91 ÷ 7 = 13. Not prime.

   You only need to test prime divisors, and only up to the square root of the number. For 91 that means 2, 3, 5 and 7 — four tests rather than eighty-nine. Students who test every number find the same answer far more slowly and give up on larger cases.

3. 3

   Build a factor tree

   60 → 6 × 10 → (2 × 3) × (2 × 5)

   Split the number into any factor pair, then keep splitting until every branch ends on a prime. The starting split does not matter — 60 as 6 × 10 or as 4 × 15 both terminate at the same primes, which is the fundamental theorem of arithmetic doing its work.

4. 4

   Write the prime factorisation in index form

   60 = 2² × 3 × 5

   Collect repeated primes into powers and write them in ascending order. Index form is the format every later question expects, and it is what makes two factorisations quick to compare when finding an HCF or LCM.

5. 5

   Check by multiplying back

   2² × 3 × 5 = 4 × 15 = 60 ✓

   The primes must multiply to the original number. This takes seconds and catches the commonest error in factor trees, which is stopping a branch on a composite number such as 9 or 15 and treating it as prime.

Why 1 is not a prime number

A prime has exactly two distinct factors. One has only itself, so it fails the definition. This looks like a technicality invented to be annoying, and it is not — it is what makes prime factorisation unique. If 1 were prime, then 6 could be written as 2 × 3, or 1 × 2 × 3, or 1 × 1 × 2 × 3, endlessly.

That uniqueness is the whole reason prime factorisation is useful. Every HCF and LCM method, and a great deal of cryptography, depends on each number having exactly one prime fingerprint. Excluding 1 is what buys that.

• Factor: divides exactly, no remainder
• Prime: exactly two factors — itself and 1
• 1 has one factor, so it is not prime
• 2 is the only even prime
• Every number above 1 has a unique prime factorisation

Two is the only even prime

Every other even number has 2 as a factor in addition to itself and 1, which disqualifies it immediately. So 2 is the only even prime, and every prime after it is odd. Students routinely assume 2 is not prime because it is even, which is the reverse of the actual argument.

The corollary is a useful shortcut: after checking 2, you never need to test another even divisor. That halves the work in any primality test.

What prime factorisation is actually for

On its own, breaking 60 into 2² × 3 × 5 answers no question anyone asked. Its value is entirely downstream: highest common factor, lowest common multiple, simplifying surds, adding fractions with awkward denominators, and simplifying algebraic fractions all become mechanical once the factorisations are in front of you.

We teach it as a tool with named uses rather than as an isolated skill, because students who cannot see what it is for practise it reluctantly and forget it quickly.

How we teach primes and factors

We teach the paired-listing method for factors because it is self-checking. A student who lists in pairs knows when the list is complete; a student listing from memory does not, and an incomplete factor list produces a wrong HCF with working that looks entirely correct.

We also spend a few minutes on why 1 is excluded. It takes almost no time and it converts an arbitrary-seeming rule into something a student can reconstruct, which matters because this is exactly the kind of definition that gets misremembered under exam pressure.