Every composite number is a product of primes, and that product is the same however the number is split, apart from order. This is the Fundamental Theorem of Arithmetic.
A factor of a whole number is a whole number that divides it with no remainder. Every whole number greater than 1 is one of two kinds.
A prime has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are primes. A composite number has more than two factors. 6 is composite, with factors 1, 2, 3 and 6. The number 1 has only one factor, so it is neither prime nor composite.
Primes are sometimes compared to atoms. The comparison holds in one way: a prime cannot be split into smaller whole-number factors, and every other whole number is built from primes. It breaks down in another way: primes combine only by multiplication, while atoms join by chemical bonds in many arrangements.
Multiplying primes, with repeats allowed, produces composite numbers: 2 × 7 = 14, 3 × 7 = 21, 2 × 2 × 3 = 12.
The reverse question is more important. Can every composite number be written as a product of primes?
Break 60 into prime factors step by step. Could there be a composite number that cannot be broken into primes?
60 = 6 × 10. Then 6 = 2 × 3 and 10 = 2 × 5. So 60 = 2 × 3 × 2 × 5 = 2² × 3 × 5.
At each step, a composite number is split into two smaller factors, each greater than 1. Each factor is either prime, and stops, or composite, and is split again. The factors get smaller at every step, and a whole number cannot get smaller forever. So the process must end, and it can only end at primes.
Therefore every composite number is a product of primes.
A factor tree records this process. Write the number at the top. Split it into any two factors. Split each composite factor again, until every branch ends in a prime.
The ends of the branches are primes. Multiplying them together gives back the number at the top: 2 × 2 × 3 × 5 = 60.
60 can also be split first as 4 × 15, or as 2 × 30. The trees look different. In every case, the primes at the ends are 2, 2, 3 and 5.
The Fundamental Theorem of Arithmetic states: every composite number can be written as a product of primes, and this product is unique apart from the order of the factors. No other collection of primes multiplies to give that number.
The existence part was shown above. The uniqueness part needs a longer proof, which uses the fact that if a prime divides a product of two numbers, it divides at least one of them. In this course the uniqueness is accepted and used.
Why does it matter that the prime factorisation is unique, and not only that one exists?
If a number had two different prime factorisations, a fact read from one of them might be contradicted by the other.
Because there is only one, any property that depends on the primes is a property of the number itself. In the next sessions this is used to find the HCF and LCM, to decide whether a fraction’s decimal ends, and to prove that √2 is irrational.
Can 4ⁿ end in the digit 0, for any natural number n?
A number ending in 0 is divisible by 10 = 2 × 5. So its prime factorisation must contain the prime 5. But 4ⁿ = (2²)ⁿ = 2²ⁿ, which contains only the prime 2. By uniqueness, it has no other prime factorisation, so it can never contain a 5. Therefore 4ⁿ never ends in 0.
No values of n had to be tried. The prime factorisation decided the question for every n at once.
1A prime number is a whole number greater than 1 that…
2Write 140 as a product of its prime factors. 140 = ____.
3The Fundamental Theorem of Arithmetic states that every composite number…
460 is factorised once starting with 6 × 10 and once starting with 4 × 15. What do the two factor trees have in common?
5Why can 4ⁿ never end in the digit 0?
6Why is 7 × 11 × 13 + 13 composite?
Next: finding the HCF and LCM from prime factorisations.
Noescio