Prime Factorisation
Prime numbers, factor trees, indices, and applications · Grade 4–6.
This booklet helps you learn about factors, step by step.
What’s a factor?
A factor is a number that fits exactly into another number — with nothing left over.
Example: 12 ÷ 3 = 4. Three fits exactly into twelve, so 3 is a factor of 12.
The factors of 12 are: 1, 2, 3, 4, 6, 12.
What you’ll learn
By the end of this paper you will be able to recognise primes, write a number as a product of its prime factors, use index notation, reason with perfect squares and cubes, and read structural facts — such as the number of factors and the biggest odd factor — straight from a factorisation. Each section opens with its own learning objective and success criteria — check these before you start the section and again when you finish. HCF and LCM are covered in the separate LCM and HCF paper.
How to use the booklet
There are seven short sections (A–G). Sections A–D are calculation questions: recognise primes, factor trees, index notation, applications. Sections E–G are word problems, easier to harder.
If you get stuck, that’s OK — stop there. The section you stop at tells your tutor exactly where you need help.
For pupils who learn best with physical things to handle:
- Plain paper for factor trees — pupil draws one tree alongside reading the lesson. Tutor draws the first; pupil draws the next.
- Two coloured highlighters — one colour for each prime in the factorisation. Helps the eye see which primes repeat.
- A small calculator (Section C onward) — only used to verify (e.g. checking 2³ × 3² × 5 = 360). Pupil writes out the working by hand first.
- A 5-minute break planned between Section D (calculation) and Section E (word problems). Switching modes is hard.
· Section A — Recognising primes. Decide whether a number is prime.
· Section B — Factor trees. Break a number down into its prime factors.
· Section C — Index notation. Evaluate basic powers such as 2³ and 3².
· Section D — Applications. Perfect squares, perfect cubes, smallest multipliers.
· Section E — Real-life applications. Word problems using prime factorisation.
· Section F — Squares and cubes in context. Word problems with perfect squares and cubes.
· Section G — Structural reasoning. Biggest odd factor and other structural questions.
Prime factors, factor trees, and index form
Aim: Write a whole number as a product of its prime factors, then in index form using powers like 2³.
A prime factor is simply a factor that is also a prime number. The clever part is that every whole number bigger than 1 can be built by multiplying prime factors together — and that product is called its prime factorisation.
· 12 = 2 × 2 × 3
· 30 = 2 × 3 × 5
· 100 = 2 × 2 × 5 × 5
Notice that each factorisation contains only prime numbers, multiplied together.
The factor tree method:
- Start with the number at the top.
- Split it into any factor pair (two numbers that multiply to give it).
- If a branch ends in a prime, circle it and stop that branch.
- If a branch ends in a composite, split it again.
- When every branch ends in a prime, multiply the leaves together — that’s the prime factorisation.
Let’s build the tree for 60 step by step, then look at the finished result.
- 60 = 2 × 30.
- 2 is prime. Stop that branch.
- Split 30: 30 = 2 × 15.
- 2 is prime. Split 15.
- 15 = 3 × 5. Both prime — stop.
- Collect leaves: 2, 2, 3, 5.
- So 60 = 2 × 2 × 3 × 5.
Drawn out, that tree looks like this — every leaf is a prime, and the 2 that appears twice is what becomes 2² in a moment:
Index form is a shorthand for repeated multiplication. A power is the small raised number you see in something like 23. It tells you how many copies of the base number to multiply together. The power is NOT a normal multiplier.
· 23 means 2 × 2 × 2 = 8. Read as “two cubed” or “two to the power of three”.
· 52 means 5 × 5 = 25. Read as “five squared”.
· 24 means 2 × 2 × 2 × 2 = 16. Read as “two to the power of four”.
23 is NOT 2 × 3 = 6.
23 IS 2 × 2 × 2 = 8.
The raised number tells you HOW MANY copies of the base to multiply together, not what to multiply the base BY.
- From the factor tree: 60 = 2 × 2 × 3 × 5.
- 2 appears twice — write 22.
- 3 appears once — just 3 (no power needed).
- 5 appears once — just 5.
- So 60 = 22 × 3 × 5.
Try one
Find the prime factorisation of 48 in index form.
Show answer
Perfect squares and cubes from prime factorisation
Aim: Decide whether a number is a perfect square or cube using its prime factorisation. Find the smallest multiplier that turns a number into a perfect square or cube.
By now you can break any number into its prime factors with a factor tree. That factorisation is the key to this lesson: it shows at a glance whether a number is a perfect square or a perfect cube.
A perfect square is an integer you get by squaring an integer: 1, 4, 9, 16, 25, 36, 49, … A perfect cube is one you get by cubing: 1, 8, 27, 64, 125, 216, …
A number is a perfect square if and only if every prime in its factorisation has an EVEN power.
A number is a perfect cube if and only if every prime power is a MULTIPLE OF 3.
- Prime factorise: 144 = 16 × 9 = 24 × 32.
- Power of 2 is 4 (even). Power of 3 is 2 (even).
- Both powers are even — so 144 IS a perfect square.
- Check: 144 = 122. ✓
Suppose n = pa × qb. Then √n = pa/2 × qb/2.
For √n to be a whole number, the halved powers a/2 and b/2 must be whole numbers — meaning a and b must be even.
Same logic for cubes: ∛n is whole only when every prime power divides cleanly by 3.
- Prime factorise 18: 18 = 2 × 32.
- For 18k to be a perfect square, every prime power must be EVEN.
- Power of 2 is 1 (odd) — need one more 2 to bump to 22.
- Power of 3 is 2 (even) — fine, leave alone.
- So k must contribute exactly one 2: k = 2.
- Check: 18 × 2 = 36 = 62. ✓
- Prime factorise 24: 24 = 23 × 3.
- For 24k to be a perfect cube, every prime power must be a MULTIPLE OF 3.
- Power of 2 is 3 (already a multiple of 3) — fine.
- Power of 3 is 1 — need to bump to 3. Add two more 3s.
- So k = 32 = 9.
- Check: 24 × 9 = 216 = 63. ✓
Try one
Find the smallest k > 0 such that 50k is a perfect square.
Show answer
Reading facts straight from a prime factorisation
Aim: Once a number is written as a product of primes, read off useful facts — how many factors it has, its biggest odd factor, and what it divides — without working out the whole number.
A prime factorisation is like a number’s blueprint. Earlier you used it to spot perfect squares and cubes; here you use the same blueprint to answer questions that would be slow to do by hand.
Skill 1 — How many factors? Every factor is built by choosing how many of each prime to include. If a prime appears to the power a, you may take 0, 1, 2, … up to a of it — that is a + 1 choices. Multiply the choices across all the primes.
- The 2 can appear 0, 1 or 2 times — 3 choices.
- The 3 can appear 0, 1 or 2 times — 3 choices.
- The 5 can appear 0 or 1 times — 2 choices.
- Multiply the choices: 3 × 3 × 2 = 18 factors.
Skill 2 — The biggest odd factor. Any factor that contains a 2 is even. So to find the biggest odd factor, simply drop every 2 and keep all the rest.
- The 2s are the only thing that can make a factor even — drop all three of them.
- What remains is 3² = 9.
- So the biggest odd factor is 9 — found without ever working out that the number is 72.
Skill 3 — Does one number divide another? One number divides another exactly when every prime power in the first is no bigger than the matching power in the second.
- Write both in prime form: 12 = 2² × 3, 360 = 2³ × 3² × 5.
- Power of 2: the 2² in 12 fits inside the 2³ in 360. ✓
- Power of 3: the 3¹ in 12 fits inside the 3² in 360. ✓
- Every prime in 12 fits, so yes — 12 divides 360 (and 360 ÷ 12 = 30).
Use (a + 1), not a. The “+ 1” is there because a prime can also appear zero times in a factor.
For 2² × 3² × 5 the answer is 3 × 3 × 2 = 18, not 2 × 2 × 1 = 4. Always multiply the (power + 1) values, never the powers themselves.
Try one
A number is 24 × 7. (a) How many factors does it have? (b) What is its biggest odd factor?
Show answer
- Factor
- A whole number that divides another exactly. 3 is a factor of 12.
- Prime number
- A whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …
- Composite number
- A whole number with more than 2 factors. 12 is composite. (1 is neither prime nor composite.)
- Prime factor
- A factor that is itself prime. The prime factors of 12 are 2 and 3.
- Prime factorisation
- A number written as a product of its prime factors. 12 = 22 × 3.
- Factor tree
- A diagram for finding the prime factorisation. Split, keep splitting, stop when every branch is prime.
- Index form (or power form)
- A shorthand for repeated multiplication. 23 means 2 × 2 × 2 = 8.
- Perfect square
- A whole number you get by squaring an integer: 1, 4, 9, 16, 25, … In prime factorisation, every prime power is EVEN.
- Perfect cube
- A whole number you get by cubing an integer: 1, 8, 27, 64, … In prime factorisation, every prime power is a MULTIPLE OF 3.
Recap before you start the paper
- Prime number = exactly 2 factors (1 and itself). 1 is NOT prime.
- Factor tree = split, split again, stop at primes.
- Prime factorisation = the number written as a product of primes (unique for every whole number).
- Index form: 23 means 2 × 2 × 2 = 8 (NOT 2 × 3 = 6).
- Perfect square from prime factorisation: every prime power is EVEN.
- Perfect cube from prime factorisation: every prime power is a MULTIPLE OF 3.
- Number of factors of pa × qb × …: (a+1)(b+1)…
- Biggest odd factor of pa × qb × …: drop the 2s.
Now try the practice questions
35 questions across seven sections. No time pressure when you’re ready. The step-by-step walkthroughs unlock after you mark.
Instructions
- Type each numeric answer in the box. For sign questions, use a minus sign (e.g.
-7). For radio questions, tap your choice. - Work through the questions carefully. There is no time pressure.
- Your tutor is looking for understanding, method, and the point where you begin to struggle.
- When finished, click Mark My Work. Your answers will then be locked, the correct answers will appear under each wrong question with the working, and you will get 5 similar practice questions inline for each one you missed.
No time pressure
Work carefully. Your tutor is looking for understanding, method, and the point where you begin to struggle.
Decide whether a number is prime.
Recognise prime numbers.
- list the first ten primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)
- decide whether a given number is prime by checking its factors
- explain why 1 is not prime and why 2 is the only even prime
Write a number as a product of its prime factors using a factor tree.
Build a factor tree and write a number as a product of its prime factors.
- split a composite number into a factor pair
- keep splitting until every branch ends at a prime
- write the prime factorisation as a multiplication (e.g. 12 = 2 × 2 × 3)
- check my answer by multiplying the primes back together
Evaluate basic powers. Convert between prime factorisation and the number.
Evaluate basic index notation and use it to write prime factorisations.
- work out a power correctly (e.g. 2³ = 8, NOT 2 × 3)
- rewrite repeated prime factors using powers (e.g. 2 × 2 × 2 × 3 = 2³ × 3)
- convert between index form and the number it represents
Perfect squares and cubes. Smallest-k multipliers.
Identify perfect squares and perfect cubes from prime factorisation.
- decide if a number is a perfect square by checking that every prime power is EVEN
- decide if a number is a perfect cube by checking that every prime power is a MULTIPLE OF 3
- find the smallest k that makes nk a perfect square or cube
Word problems: packing, grouping, splitting equally.
Apply prime factorisation to real-life contexts.
- recognise when a real-life problem needs factor-based reasoning
- use prime factorisation to find how groups, packs or rows divide evenly
- check that the answer makes sense in context
Word problems with perfect squares and cubes.
Apply perfect-square and perfect-cube reasoning to word problems (LO 4 in context).
- spot when a word problem requires a perfect square or cube
- use prime factorisation to find a missing side or count
- check by squaring or cubing back
Biggest odd factor and other structural questions.
Use the structure of a prime factorisation to reason without doing big calculations.
- find the biggest odd factor of a number by dropping the 2s
- decide whether one number is divisible by another using prime factorisations
- count how many factors a number has by listing systematically