Primes & unique factorisation
A prime has exactly two positive divisors, which is why 1 is not prime — treating it as prime would break unique factorisation. The fundamental theorem of arithmetic says every integer above 1 factors into primes in exactly one way up to order, and that uniqueness is what makes gcd, lcm and divisor counting computable from exponents.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Test primality by trial division to √nIf n has no prime factor at or below √n, it is prime — any factorisation would have a factor that small.
- Factorise by dividing out small primesRepeatedly divide by 2, then 3, 5, 7 … until the quotient is 1.
- Read divisor facts off the exponentsFor n = p₁^a₁…p_k^a_k, the number of divisors is Π(aᵢ+1).
- Compute gcd and lcm from exponentsgcd takes the minimum exponent of each prime, lcm the maximum.
Worked example
Factorise 360 and find how many positive divisors it has.
- 360 = 2 × 180 = 4 × 90 = 8 × 45.
- 45 = 9 × 5 = 3² × 5.
- So 360 = 2³ × 3² × 5¹.
- Divisor count: (3+1)(2+1)(1+1).
Answer. 360 = 2³·3²·5, with 4 × 3 × 2 = 24 positive divisors.
Where marks get dropped
These are the specific errors that cost credit on primes & unique factorisation questions — QED's rubric penalises each of them separately.
- Calling 1 prime. It has only one divisor, and admitting it would destroy the uniqueness of factorisation.
- Trial dividing past √n. Once you exceed the square root, any remaining factor would pair with a smaller one already tested.
- Forgetting exponent +1 in the divisor formula — the exponent 0 is a legitimate choice for each prime.
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Primes & unique factorisation — frequently asked questions
Why is 1 not prime?
Because unique factorisation would fail: 6 = 2·3 = 1·2·3 = 1·1·2·3 and so on. Excluding 1 keeps the factorisation unique.
How many primes are there?
Infinitely many. Euclid’s proof: given any finite list, p₁…p_n + 1 has a prime factor not on the list.
Why stop trial division at √n?
If n = ab with both factors above √n then ab > n, a contradiction. So the smaller factor is at most √n.
The rest of Number Theory
Divisibility, primes, gcd, modular arithmetic. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Divisibility & the division algorithm
- 2Primes & unique factorisation
- 3gcd, lcm & the Euclidean algorithm
- 4Modular arithmetic & congruences
- 5Solving linear congruences
- 6Fermat’s little theorem
- 7The Chinese remainder theorem
- 8Modular inverses & fast exponentiation
- 9Euler’s totient φ(n) & Euler’s theorem
- 10Base-b representations & digit arguments
- 11RSA & modular exponentiation in cryptography
- 12Wilson’s theorem & primality testing
- 13Linear Diophantine equations ax + by = c
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