SPEC N4, N5 • NUMBER
⭐ HIGH PRIORITY
Primes, Factors & Multiples
HCF, LCM & Prime Factor Decomposition. Factor trees, index notation, and the one sanity check that stops you swapping HCF and LCM ever again.
Key Definitions
Five terms that appear on every GCSE paper. Learn them in mark-scheme language.
Prime number
A positive integer greater than 1 with exactly two distinct factors: 1 and itself. 2 is the only even prime.
Factor
An integer that divides another integer with no remainder. Factors of 12: 1, 2, 3, 4, 6, 12.
Multiple
A product of a given integer and any other integer. Multiples of 4: 4, 8, 12, 16, 20, …
HCF & LCM
HCF (Highest Common Factor): the largest integer that is a factor of all given numbers. LCM (Lowest Common Multiple): the smallest positive integer that is a multiple of all given numbers.
Why 1 is Not Prime
Students often list 1 as the first prime. The primes start at 2.
The logic
A prime has exactly two distinct factors: 1 and itself. The word "distinct" matters.
1 has only one factor (1). That's not two → not prime.
2 has factors 1 and 2 → two distinct factors → prime.
7 has factors 1 and 7 → two distinct factors → prime.
Classification
Every positive integer above 1 is either prime or composite (built from primes). The number 1 sits outside this — it is neither prime nor composite.
Odd ≠ prime
9 is odd but 9 = 3 × 3 → not prime. 15, 21, 25, 27, 33 are all odd but composite. Always check by trying to divide by small primes.
Primes Under 50 — Memorise These
Fifteen primes under 50. On Paper 1 you can't look them up. Commit to memory.
Checking if a larger number is prime
Try dividing by each prime in order: 2, 3, 5, 7, 11, 13, … Stop once the divisor squared exceeds the number. For 97: √97 ≈ 9.8, so you only need to test up to 7. 97 ÷ 2 fails, ÷ 3 fails, ÷ 5 fails, ÷ 7 fails → 97 is prime.
Prime Factor Decomposition — Factor Tree
Every integer above 1 can be written as a product of primes in exactly one way. The factor tree finds it.
Method
Pick any two factors. It doesn't matter which — the final answer is unique.
Branch any non-prime further.
Stop when every leaf is prime (highlighted here in blue).
Write the product in index notation.
Why Index Notation Matters
AQA mark schemes explicitly credit index form. Writing out every prime individually can lose you a mark even though the factorisation is correct.
B2 only
2 × 2 × 2 × 3 × 5 × 5
All primes correct. But not in index form → loses the third mark.
B3 full
2³ × 3 × 5²
Same factorisation, in index form → full marks.
Mark scheme language
From a real AQA mark scheme (Jun24 P3 Q17): "a and b identified as 2 and 5, written with correct indices." A student who writes 5³ × 2² instead of 2³ × 5² explicitly scores B0 — the indices must match the base correctly.
HCF — Lowest Powers
Once both numbers are in prime factor form, HCF is mechanical.
Rule
HCF = take the lowest power of each prime that appears in both decompositions. Multiply them together. Primes appearing in only one of the numbers are excluded.
Example: HCF of 12 and 18
12 = 2² × 318 = 2 × 3²
Lowest power of 2: min(2, 1) = 2¹
Lowest power of 3: min(1, 2) = 3¹
HCF = 2 × 3 = 6
Sanity check
The HCF is never larger than either original number. If your HCF comes out bigger than one of them, you've computed the LCM by mistake.
LCM — Highest Powers
The mirror of HCF. Same setup, opposite rule.
Rule
LCM = take the highest power of each prime that appears in any decomposition. Multiply them together. Include every prime, even if only in one number.
Example: LCM of 12 and 18
12 = 2² × 318 = 2 × 3²
Highest power of 2: max(2, 1) = 2²
Highest power of 3: max(1, 2) = 3²
LCM = 2² × 3² = 4 × 9 = 36
Sanity check
The LCM is never smaller than either original number. If your LCM comes out smaller than one of them, you've computed the HCF by mistake.
MCE Worked Example 4 MARKS01
Find the HCF and LCM of 72 and 120.
•
MMethod. Write each number as a product of prime factors. HCF = lowest power of each shared prime. LCM = highest power of each prime appearing in either.
•
CCalculate.
72 = 8 × 9 = 2³ × 3²
120 = 8 × 15 = 2³ × 3 × 5
HCF (lowest powers):
2: min(3, 3) = 2³; 3: min(2, 1) = 3¹; 5: not in 72
HCF = 2³ × 3 = 8 × 3 = 24
LCM (highest powers):
2: max(3, 3) = 2³; 3: max(2, 1) = 3²; 5: 5¹
LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360
•
EEvaluate.HCF = 24, LCM = 360.
Check: HCF × LCM = 24 × 360 = 8640. And 72 × 120 = 8640. ✔
Mark scheme: M1 for each factorisation (2 marks); A1 for HCF; A1 for LCM.
The HCF × LCM Sanity Check
One of the most useful facts in GCSE number theory — and one students never get told explicitly.
HCF(a, b) × LCM(a, b) = a × b
Why it works
Every prime in a or b appears once in the HCF (its lowest power) and once in the LCM (its highest power). Multiplying them adds the powers, reproducing the full product of the two numbers.
Example — 72 & 120
HCF × LCM = 24 × 360 = 8640
a × b = 72 × 120 = 8640 ✔
How to use it in the exam
If you compute the HCF and LCM on a 4-mark question, check by multiplying them and comparing to the product of the original numbers. If they match, you're done. If they don't — one of your values is wrong.
Weak vs Strong: Is 9 Prime?
A very common opening mistake. Students see odd and think prime.
List the prime numbers below 12.
Weak student
"Odd numbers are prime."
2, 3, 5, 7, 9, 11
9 = 3 × 3. Has three factors (1, 3, 9) → not prime. Mark lost.
Strong student
"Check each: does it divide by 2, 3, 5, 7?"
2, 3, 5, 7, 11
9 ÷ 3 = 3 exactly → composite. Excluded.
Other common traps
Students often list 27 as prime (27 = 3³). Also watch for 15, 21, 25, 33, 35, 39, 49 — all odd, all composite. Flagged in AQA examiner reports (Jun23 P1 Q5).
Examiner Alert: HCF/LCM Reversal
What examiners report
Students swap the two. They write HCF when asked for LCM, or vice versa. Often the working is completely correct — the student just copies the wrong value onto the answer line.
The sanity-check rule
HCF ≤ both numbers. If your "HCF" is bigger than either original number, you've given the LCM.
LCM ≥ both numbers. If your "LCM" is smaller than either original, you've given the HCF.
Memory hook
Highest Common Factor → "high" sounds big, but a factor is small. HCF is small. LCM is big.
Examiner Alert: 1 is Not Prime
What examiners see
Students list 1 as the first prime. This loses a mark on "list the primes up to n" type questions, and breaks prime factorisation arguments (because 1 × 1 × 1 × anything is a valid but meaningless factorisation).
What to write instead
The primes start at 2. The number 1 is neither prime nor composite — it sits in its own category. In any prime factor decomposition, do not write 1 as a factor.
Quick reference
2 is the only even prime. All other even numbers have 2 as a factor, so they're composite.
Common Errors
Students often writeCorrect versionWhy it matters
9 listed as prime9 = 3 × 3 (composite)Foundational classification error
HCF of 12 and 18 = 3636 is the LCM; HCF = 6Reversal costs both marks
Prime factorisation without indices: 2×2×3×52² × 3 × 5Final B-mark often reserved for index notation
1 listed as primePrimes start at 2Definition error; invalidates factorisation argument
27 listed as prime27 = 3³ (composite)Flagged in AQA Jun23 examiner report
Practice Question 1
Express 108 as a product of its prime factors. Give your answer in index form.3 MARKS
1
Build a factor tree. Start with 108 → 2 × 54.
2
54 → 2 × 27. 27 → 3 × 9. 9 → 3 × 3.
3
Leaves (all prime): 2, 2, 3, 3, 3
4
Write in index form: 108 = 2² × 3³
Answer: 108 = 2² × 3³
Check: 2² × 3³ = 4 × 27 = 108 ✔
Mark scheme: B1 for full factor tree with all primes, B1 for complete product, B1 for index form.
Practice Question 2
You are given that 108 = 2² × 3³ and 24 = 2³ × 3.
(a) Find the HCF of 108 and 24. 2 MARKS
(b) Find the LCM of 108 and 24. 2 MARKS
(a)
HCF — lowest powers.
Lowest power of 2: min(2, 3) = 2² = 4
Lowest power of 3: min(3, 1) = 3¹ = 3
HCF = 4 × 3 = 12
(b)
LCM — highest powers.
Highest power of 2: max(2, 3) = 2³ = 8
Highest power of 3: max(3, 1) = 3³ = 27
LCM = 8 × 27 = 216
Answers: HCF = 12, LCM = 216
Check: HCF × LCM = 12 × 216 = 2592. And 108 × 24 = 2592 ✔
Practice Question 3
Two buses leave a station at 8:00 a.m.
Bus A departs every 12 minutes. Bus B departs every 18 minutes.
At what time do they next leave the station together?3 MARKS
1
The buses leave together at the lowest common multiple of their intervals → find LCM(12, 18).
2
Factorise:
12 = 2² × 3
18 = 2 × 3²
3
LCM (highest powers):
LCM = 2² × 3² = 4 × 9 = 36 minutes
4
Add to 8:00 a.m. → 8:36 a.m.
Answer: 8:36 a.m.
Interpretation mark (AO3): don't stop at "36 minutes" — the question asks for a time. Converting to 8:36 a.m. is where the final mark is awarded.
Summary
Definitions
Prime: exactly two distinct factors (1 and itself). 2 is the only even prime. 1 is NOT prime.
Factor: divides with no remainder.
Multiple: product of the number and any integer.
HCF: largest common factor. Always ≤ both numbers.
LCM: smallest common multiple. Always ≥ both numbers.
Primes Under 50 (15 of them)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
HCF and LCM from Prime Factors
HCF: lowest power of each shared prime
LCM: highest power of each prime in either number
Always write factorisations in index notation — AQA credits this
The Sanity Check
HCF × LCM = a × b
If they don't match, one of your values is wrong.
Ready for 1.04
Next: Powers, Roots & Index Laws — the seven index laws, negative and fractional indices, and why 2⁻³ is not a negative number.
Next: Key Definitions