Prime Factorisation
Every whole number is built from primes, the way every molecule is built from atoms. Learn to take a number apart, write down its recipe, and use that recipe to find HCFs and LCMs without guessing.
National curriculum KS3: prime numbers, prime factorisation (including product notation and index form), HCF and LCM1 · What is a prime?
A factor of a number divides into it exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. A prime is a whole number greater than 1 whose only factors are 1 and itself. 7 is prime: nothing except 1 and 7 goes into it. 9 is not: 3 × 3 = 9.
Here's a way to see it. Try to arrange a number of dots into a neat rectangle (more than one row and more than one column). If you can, the number is composite. If the only arrangement is a single line, it's prime.
Why isn't 1 prime? Because the whole point of primes is that every number has exactly one prime recipe. If 1 counted, then 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3 … and the recipe would never be unique. So by agreement 1 is neither prime nor composite. 2 is the only even prime.
2 · Finding the primes: the sieve
Eratosthenes, a Greek librarian around 240 BC, found a way to filter out all the primes up to any limit with no division at all. Write out the numbers. Take the first unmarked number (2), keep it, and cross out all its multiples. Move to the next unmarked number (3), keep it, cross out its multiples. Repeat. What survives is prime.
There are 25 primes below 100. It's worth knowing the first few by heart: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. They are the building blocks for everything that follows.
3 · Taking a number apart: factor trees
Any composite number can be split into two factors. If a factor is prime, circle it and stop. If it isn't, split it again. Keep going until every branch ends in a prime. The primes at the ends are the number's prime factors.
Build one yourself. Type a number, then click any orange box to split it. You choose the factor pair. Try splitting the same number two different ways and compare what you end up with.
However you split, you always end with the same collection of primes. That's the Fundamental Theorem of Arithmetic: every whole number above 1 has exactly one prime factorisation (ignoring order). 60 is always 2 × 2 × 3 × 5, whether you started with 6 × 10 or 4 × 15 or 2 × 30.
Writing it in index form
Repeated primes get collected using powers: 2 × 2 × 3 × 5 = 2² × 3 × 5. Index form is shorter and, as you'll see, makes HCF and LCM almost automatic. Convention: write the primes in increasing order.
The ladder method (division method)
Some people prefer repeated division to trees. Divide by the smallest prime that goes in, write the answer underneath, repeat until you reach 1. The primes down the left are the answer.
The ladder is secretly a tree as well. Each division is a node with two children: the prime you divided by on the left (a leaf, it stops there) and the quotient on the right, which gets divided again. Because every node has at most two children it is a binary tree, and because every left branch stops at once, it leans all the way to the right. The ladder on the left and the tree on the right are the same working, drawn two ways.
Compare it with the free factor tree above. The free tree lets you split any way you like and can be bushy; the division tree always peels off the smallest prime first, so it is tall and thin. Both are binary trees, both end in the same leaves. If you've met binary trees in computing, this is the same data structure: the leaves, read left to right, are the prime factorisation in increasing order.
4 · What's it for? HCF and LCM
The prime recipe is a kind of X-ray of a number. Once you can see inside two numbers, two classic problems become easy.
The highest common factor (HCF, máximo común divisor in Spanish) is the biggest number that divides into both. The lowest common multiple (LCM, mínimo común múltiplo) is the smallest number both divide into. Put the prime factors into a Venn diagram: shared primes go in the middle.
Two rules fall out of the picture:
- HCF: multiply the primes in the overlap. In index form, take the lower power of each shared prime.
- LCM: multiply every prime in the diagram once. In index form, take the higher power of each prime that appears anywhere.
And a neat check: HCF × LCM = A × B, always.
5 · Where this shows up
Fractions
Simplifying ⁶⁰⁄₈₄ means dividing top and bottom by their HCF (12) to get ⁵⁄₇. Adding ¹⁄₆ + ¹⁄₈ needs a common denominator: the LCM, 24.
Timetables and cycles
One bus every 12 minutes, another every 18. Both leave at 9:00; when do they next leave together? LCM(12, 18) = 36, so 9:36.
Tiling and packing
Largest square tile that covers a 60 cm × 84 cm floor with no cutting? HCF(60, 84) = 12 cm. Smallest square you can tile with 60 × 84 rectangles? LCM gives the side: 420 cm.
Internet security
RSA encryption, which protects card payments, multiplies two enormous primes (hundreds of digits). Multiplying is instant; factorising the result back would take the fastest computers longer than the age of the universe. That one-way gap is the lock.
Simplifying surds (GCSE)
√72: factorise 72 = 2³ × 3², pull out the square pairs, get 6√2. The factor tree is the first step every time.
Spotting squares and cubes
A number is a perfect square exactly when every power in its index form is even. 3600 = 2⁴ × 3² × 5² is square (√3600 = 2² × 3 × 5 = 60). 360 = 2³ × 3² × 5 isn't.
Worked examples
Write 504 as a product of its prime factors
Find the HCF and LCM of 72 and 120
Exam-style: is 2⁴ × 3 × 5² a multiple of 60?
Quick check
What you've learned
Primes are the numbers that can't be split. Every other whole number splits into primes in exactly one way, which you can find with a factor tree or the ladder, and write compactly in index form. Lay two factorisations in a Venn diagram and the overlap gives the HCF, the whole thing gives the LCM. Those two tools run fractions, timetables, tiling and, scaled up to numbers with hundreds of digits, the encryption behind every online payment.